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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,
By THE MACMILLAN COMPANY.
Copyright, 1924,
By EMMA B. CONANT.
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
J. S. Cushing Co.—Berwick & Smith Co.
Norwood, Mass., U.S.A.
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.
Worcester, Mass., Nov. 12, 1895.
- Chapter I.
- Counting 1
- Chapter II.
- Number System Limits 21
- Chapter III.
- Origin of Number Words 37
- Chapter IV.
- Origin of Number Words (continued) 74
- Chapter V.
- Miscellaneous Number Bases 100
- Chapter VI.
- The Quinary System 134
- Chapter VII.
- The Vigesimal System 176
- Index 211
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.
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.
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.
The process of formation indicated in 11 is used in the
succeeding numerals up to 19.
20.kwillik'yënästem'thlan= two times all the fingers.
100.ässiästem'thlak'ya= the fingers all the fingers.
1000.ässiästem'thlanak'yënästem'thla= the fingers all the fingers times all the fingers.
The only numerals calling for any special note are
those for 11 and 9. For 9 we should naturally expect
a word corresponding in structure and meaning to the
words for 7 and 8. But instead of the “four brought
to and held up with the rest,” for which we naturally
look, the Zuñi, to show that he has used all of his
fingers but one, says “all but all are held up with the
rest.” To express 11 he cannot use a similar form of
composition, since he has already used it in constructing
his word for 6, so he says “all the fingers and
another over above held.”
The one remarkable point to be noted about the
Zuñi scale is, after all, the formation of the words for
1 and 2. While the savage almost always counts on
his fingers, it does not seem at all certain that these
words would necessarily be of finger formation. The
savage can always distinguish between one object and
two objects, and it is hardly reasonable to believe that
any external aid is needed to arrive at a distinct perception
of this difference. The numerals for 1 and 2
would be the earliest to be formed in any language,
and in most, if not all, cases they would be formed
long before the need would be felt for terms to
describe any higher number. If this theory be correct,
we should expect to find finger names for numerals
beginning not lower than 3, and oftener with 5
than with any other number. The highest authority
has ventured the assertion that all numeral words have
their origin in the names of the fingers;69 substantially
the same conclusion was reached by Professor Pott, of
Halle, whose work on numeral nomenclature led him
deeply into the study of the origin of these words.
But we have abundant evidence at hand to show that,
universal as finger counting has been, finger origin for
numeral words has by no means been universal. That
it is more frequently met with than any other origin is
unquestionably true; but in many instances, which will
be more fully considered in the following chapter, we
find strictly non-digital derivations, especially in the
case of the lowest members of the scale. But in nearly
all languages the origin of the words for 1, 2, 3, and
4 are so entirely unknown that speculation respecting
them is almost useless.
An excellent illustration of the ordinary method of
formation which obtains among number scales is furnished
by the Eskimos of Point Barrow,70 who have pure
numeral words up to 5, and then begin a systematic
course of word formation from the names of their
fingers. If the names of the first five numerals are of
finger origin, they have so completely lost their original
form, or else the names of the fingers themselves have
so changed, that no resemblance is now to be detected
between them. This scale is so interesting that it is
given with considerable fulness, as follows:
1.atauzik. |
2.madro. |
3.pinasun. |
4.sisaman. |
5.tudlemut. |
6.atautyimin akbinigin [tudlimu(t)]= 5 and 1 on the next.
7.madronin akbinigin= twice on the next.
8.pinasunin akbinigin= three times on the next.
9.kodlinotaila= that which has not its 10.
10.kodlin= the upper part—i.e. the fingers.
14.akimiaxotaityuna= I have not 15.
15.akimia. [This seems to be a real numeral word.] |
20.inyuina= a man come to an end.
25.inyuina tudlimunin akbinidigin= a man come to an end and 5 on the next.
30.inyuina kodlinin akbinidigin= a man come to an end and 10 on the next.
35.inyuina akimiamin aipalin= a man come to an end accompanied by 1 fifteen times.
40.madro inyuina= 2 men come to an end.
In this scale we find the finger origin appearing so
clearly and so repeatedly that one feels some degree of
surprise at finding 5 expressed by a pure numeral instead
of by some word meaning hand or fingers of one
hand. In this respect the Eskimo dialects are somewhat
exceptional among scales built up of digital words.
The system of the Greenland Eskimos, though differing
slightly from that of their Point Barrow cousins, shows
the same peculiarity. The first ten numerals of this
scale are:71
1.atausek. |
2.mardluk. |
3.pingasut. |
4.sisamat. |
5.tatdlimat. |
6.arfinek-atausek= to the other hand 1.
7.arfinek-mardluk= to the other hand 2.
8.arfinek-pingasut= to the other hand 3.
9.arfinek-sisamat= to the other hand 4.
10.kulit. |
The same process is now repeated, only the feet instead
of the hands are used; and the completion of the
second 10 is marked by the word innuk, man. It may
be that the Eskimo word for 5 is, originally, a digital
word, but if so, the fact has not yet been detected.
From the analogy furnished by other languages we are
justified in suspecting that this may be the case; for
whenever a number system contains digital words, we
expect them to begin with five, as, for example, in the
Arawak scale,72 which runs:
1.abba. |
2.biama. |
3.kabbuhin. |
4.bibiti. |
5.abbatekkábe= 1 hand.
6.abbatiman= 1 of the other.
7.biamattiman= 2 of the other.
8.kabbuhintiman= 3 of the other.
9.bibitiman= 4 of the other.
10.biamantekábbe= 2 hands.
11.abba kutihibena= 1 from the feet.
20.abba lukku= hands feet.
The four sets of numerals just given may be regarded
as typifying one of the most common forms of
primitive counting; and the words they contain serve
as illustrations of the means which go to make up the
number scales of savage races. Frequently the finger
and toe origin of numerals is perfectly apparent, as
in the Arawak system just given, which exhibits the
simplest and clearest possible method of formation.
Another even more interesting system is that of the
Montagnais of northern Canada.
73 Here, as in the Zuñi
scale, the words are digital from the outset.
1.inl'are= the end is bent.
2.nak'e= another is bent.
3.t'are= the middle is bent.
4.dinri= there are no more except this.
5.se-sunla-re= the row on the hand.
6.elkke-t'are= 3 from each side.
7. | {t'a-ye-oyertan= there are still 3 of them. |
inl'as dinri= on one side there are 4 of them. |
8.elkke-dinri= 4 on each side.
9.inl'a-ye-oyert'an= there is still 1 more.
10.onernan= finished on each side.
11.onernan inl'are ttcharidhel= 1 complete and 1.
12.onernan nak'e ttcharidhel= 1 complete and 2, etc.
The formation of 6, 7, and 8 of this scale is somewhat
different from that ordinarily found. To express 6, the
Montagnais separates the thumb and forefinger from
the three remaining fingers of the left hand, and bringing
the thumb of the right hand close to them, says:
“3 from each side.” For 7 he either subtracts from
10, saying: “there are still 3 of them,” or he brings
the thumb and forefinger of the right hand up to the
thumb of the left, and says: “on one side there are 4
of them.” He calls 8 by the same name as many of
the other Canadian tribes, that is, two 4's; and to show
the proper number of fingers, he closes the thumb and
little finger of the right hand, and then puts the three
remaining fingers beside the thumb of the left hand.
This method is, in some of these particulars, different
from any other I have ever examined.
It often happens that the composition of numeral
words is less easily understood, and the original meanings
more difficult to recover, than in the examples
already given. But in searching for number systems
which show in the formation of their words the influence
of finger counting, it is not unusual to find those
in which the derivation from native words signifying
finger, hand, toe, foot, and man, is just as frankly obvious
as in the case of the Zuñi, the Arawak, the Eskimo,
or the Montagnais scale. Among the Tamanacs,74 one
of the numerous Indian tribes of the Orinoco, the numerals
are as strictly digital as in any of the systems
already examined. The general structure of the Tamanac
scale is shown by the following numerals:
5.amgnaitone= 1 hand complete.
6.itacono amgna pona tevinitpe= 1 on the other hand.
10.amgna aceponare= all of the 2 hands.
11.puitta pona tevinitpe= 1 on the foot.
16.itacono puitta pona tevinitpe= 1 on the other foot.
20.tevin itoto= 1 man.
21.itacono itoto jamgnar bona tevinitpe= 1 on the hands of another man.
In the Guarani
75 language of Paraguay the same
method is found, with a different form of expression
for 20. Here the numerals in question are
5.asepopetei= one hand.
10.asepomokoi= two hands.
20.asepo asepi abe= hands and feet.
Another slight variation is furnished by the Kiriri
language,
76 which is also one of the numerous South
American Indian forms of speech, where we find the
words to be
5.mi biche misa= one hand.
10.mikriba misa sai= both hands.
20.mikriba misa idecho ibi sai= both hands together with the feet.
Illustrations of this kind might be multiplied almost
indefinitely; and it is well to note that they may be
drawn from all parts of the world. South America is
peculiarly rich in native numeral words of this kind;
and, as the examples above cited show, it is the field
to which one instinctively turns when this subject is
under discussion. The Zamuco numerals are, among
others, exceedingly interesting, giving us still a new
variation in method. They 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 Maipures
78 of the Orinoco, who count
1.papita. |
2.avanume. |
3.apekiva. |
4.apekipaki. |
5.papitaerri capiti= 1 only hand.
6.papita yana pauria capiti purena= 1 of the other hand we take.
10.apanumerri capiti= 2 hands.
11.papita yana kiti purena= 1 of the toes we take.
20.papita camonee= 1 man.
40.avanume camonee= 2 men.
60.apekiva camonee= 3 men, etc.
In all the examples thus far given, 20 is expressed
either by the equivalent of “man” or by some formula
introducing the word “feet.” Both these modes of expressing
what our own ancestors termed a “score,” are
so common that one hesitates to say which is of the
more frequent use. The following scale, from one of
the Betoya 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 language
95 the relation between 2 and
7, and 3 and 8, is no less clear. Here the words are:
2.tuklo.7.untuklo. |
3.tuchina.8.untuchina. |
The Nez Percés
96 repeat the first three words of their
scale in their 6, 7, and 8 respectively, as a comparison of
these numerals will show.
1.naks.6.oilaks. |
2.lapit.7.oinapt. |
3.mitat.8.oimatat. |
In all these cases the essential point of the method
is contained in the repetition, in one way or another,
of the numerals of the second quinate, without the use
with each one of the word for 5. This may make 6,
7, 8, and 9 appear as second 1, second 2, etc., or another
1, another 2, etc.; or, more simply still, as 1 more, 2
more, etc. It is the method which was briefly discussed
in the early part of the present chapter, and is by no
means uncommon. In a decimal scale this repetition
would begin with 11 instead of 6; as in the system found
in use in Tagala and Pampanaga, two of the Philippine
Islands, where, for example, 11, 12, and 13 are:
97
11.labi-n-isa= over 1.
12.labi-n-dalaua= over 2.
13.labi-n-tatlo= over 3.
A precisely similar method of numeral building is used
by some of our Western Indian tribes. Selecting a few
of the Assiniboine numerals
98 as an illustration, we have
11.ak kai washe= more 1.
12.ak kai noom pah= more 2.
13.ak kai yam me nee= more 3.
14.ak kai to pah= more 4.
15.ak kai zap tah= more 5.
16.ak kai shak pah= more 6, etc.
A still more primitive structure is shown in the
numerals of the Mboushas
99 of Equatorial Africa. Instead
of using 5-1, 5-2, 5-3, 5-4, or 2d 1, 2d 2, 2d 3,
2d 4, in forming their numerals from 6 to 9, they proceed
in the following remarkable and, at first thought,
inexplicable manner to form their compound numerals:
1.ivoco. |
2.beba. |
3.belalo. |
4.benai. |
5.betano. |
6.ivoco beba= 1-2.
7.ivoco belalo= 1-3.
8.ivoco benai= 1-4.
9.ivoco betano= 1-5.
10.dioum. |
No explanation is given by Mr. du Chaillu for such
an apparently incomprehensible form of expression as,
for example, 1-3, for 7. Some peculiar finger pantomime
may accompany the counting, which, were it
known, would enlighten us on the Mbousha's method
of arriving at so anomalous a scale. Mere repetition
in the second quinate of the words used in the first
might readily be explained by supposing the use of fingers
absolutely indispensable as an aid to counting, and
that a certain word would have one meaning when associated
with a certain finger of the left hand, and another
meaning when associated with one of the fingers of the
right. Such scales are, if the following are correct,
actually in existence among the islands of the Pacific.
Balad.100
1.parai.
2.paroo.
3.pargen.
4.parbai.
5.panim.
6.parai.
7.paroo.
8.pargen.
9.parbai.
10.panim.
Uea.100
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 Africa
103 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 Yoruba
105 scale,
which is in many respects one of the most peculiar in
existence. Here the words for 11, 12, etc., are formed
by adding the suffix -la, great, to the words for 1, 2,
etc., thus:
1.eni, or okan. |
2.edzi. |
3.eta. |
4.erin. |
5.arun. |
6.efa. |
7.edze. |
8.edzo. |
9.esan. |
10.ewa. |
11.okanla= great 1.
12.edzila= great 2.
13.etala= great 3.
14.erinla= great 4, etc.
40.ogodzi= string.
200.igba= heap.
The word for 40 was adopted because cowrie shells,
which are used for counting, were strung by forties;
and
igba, 200, because a heap of 200 shells was five
strings, and thus formed a convenient higher unit for
reckoning. Proceeding in this curious manner,106 they
called 50 strings 1 afo or head; and to illustrate their
singular mode of reckoning—the king of the Dahomans,
having made war on the Yorubans, and attacked
their army, was repulsed and defeated with a loss of
“two heads, twenty strings, and twenty cowries” of
men, or 4820.
The number scale of the Abipones,107 one of the low
tribes of the Paraguay region, contains two genuine
curiosities, and by reason of those it deserves a place
among any collection of numeral scales designed to
exhibit the formation of this class of words. It is:
1.initara= 1 alone.
2.inoaka. |
3.inoaka yekaini= 2 and 1.
4.geyenknate= toes of an ostrich.
5.neenhalek= a five coloured, spotted hide,
or hanambegen= fingers of 1 hand. |
10.lanamrihegem= fingers of both hands.
20.lanamrihegem cat gracherhaka anamichirihegem = fingers of both hands together with toes of both feet. |
That the number sense of the Abipones is but little,
if at all, above that of the native Australian tribes, is
shown by their expressing 3 by the combination 2
and 1. This limitation, as we have already seen, is
shared by the Botocudos, the Chiquitos, and many of
the other native races of South America. But the
Abipones, in seeking for words with which to enable
themselves to pass beyond the limit 3, invented the
singular terms just given for 4 and 5. The ostrich,
having three toes in front and one behind on each foot
presented them with a living example of 3 + 1; hence
“toes of an ostrich” became their numeral for 4. Similarly,
the number of colours in a certain hide being five,
the name for that hide was adopted as their next
numeral. At this point they began to resort to digital
numeration also; and any higher number is expressed
by that method.
In the sense in which the word is defined by mathematicians,
number is a pure, abstract concept. But a
moment's reflection will show that, as it originates
among savage races, number is, and from the limitations
of their intellect must be, entirely concrete. An abstract
conception is something quite foreign to the essentially
primitive mind, as missionaries and explorers have found
to their chagrin. The savage can form no mental concept
of what civilized man means by such a word as
“soul”; nor would his idea of the abstract number 5 be
much clearer. When he says five, he uses, in many cases
at least, the same word that serves him when he wishes
to say hand; and his mental concept when he says five
is of a hand. The concrete idea of a closed fist or an
open hand with outstretched fingers, is what is upper-most
in his mind. He knows no more and cares no
more about the pure number 5 than he does about the
law of the conservation of energy. He sees in his
mental picture only the real, material image, and his
only comprehension of the number is, “these objects
are as many as the fingers on my hand.” Then, in
the lapse of the long interval of centuries which intervene
between lowest barbarism and highest civilization,
the abstract and the concrete become slowly dissociated,
the one from the other. First the actual hand picture
fades away, and the number is recognized without the
original assistance furnished by the derivation of the
word. But the number is still for a long time a certain
number of objects, and not an independent concept.
It is only when the savage ceases to be wholly an
animal, and becomes a thinking human being, that
number in the abstract can come within the grasp of
his mind. It is at this point that mere reckoning
ceases, and arithmetic begins.
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”:
One's none,
Two's some,
Three's a many,
Four's a penny,
Five's a little hundred.
125
Any real numeral formation by the use of “little,”
with the name of some higher unit, would, of course,
be impossible. The numeral scale must be complete
before the nursery rhyme can be manufactured.
It is not to be supposed from the observations that
have been made on the formation of savage numeral
scales that all, or even the majority of tribes, proceed
in the awkward and faltering manner indicated by
many of the examples quoted. Some of the North
American Indian tribes have numeral scales which
are, as far as they go, as regular and almost as simple
as our own. But where digital numeration is extensively
resorted to, the expressions for higher numbers
are likely to become complex, and to act as a real bar
to the extension of the system. The same thing is
true, to an even greater degree, of tribes whose number
sense is so defective that they begin almost from
the outset to use combinations. If a savage expresses
the number 3 by the combination 2-1, it will at once
be suspected that his numerals will, by the time he
reaches 10 or 20, become so complex and confused that
numbers as high as these will be expressed by finger
pantomime rather than by words. Such is often the
case; and the comment is frequently made by explorers
that the tribes they have visited have no words for
numbers higher than 3, 4, 5, 10, or 20, but that counting
is carried beyond that point by the aid of fingers
or other objects. So reluctant, in many cases, are savages
to count by words, that limits have been assigned
for spoken numerals, which subsequent investigation
proved to fall far short of the real extent of the number
systems to which they belonged. One of the south-western
Indian tribes of the United States, the Comanches,
was for a time supposed to have no numeral
words below 10, but to count solely by the use of
fingers. But the entire scale of this taciturn tribe was
afterward discovered and published.
To illustrate the awkward and inconvenient forms of
expression which abound in primitive numeral nomenclature,
one has only to draw from such scales as those
of the Zuñi, or the Point Barrow Eskimos, given in the
last chapter. Terms such as are found there may
readily be duplicated from almost any quarter of the
globe. The Soussous of Sierra Leone126 call 99 tongo
solo manani nun solo manani, i.e. to take (10 understood)
5 + 4 times and 5 + 4. The Malagasy expression
for 1832 is127 roambistelo polo amby valonjato amby
arivo, 2 + 30 + 800 + 1000. The Aztec equivalent for 399
is128 caxtolli onnauh poalli ipan caxtolli onnaui, (15 + 4)
× 20 + 15 + 4; and the Sioux require for 29 the ponderous
combination129 wick a chimen ne nompah sam pah
nep e chu wink a. These terms, long and awkward as
they seem, are only the legitimate results which arise
from combining the names of the higher and lower
numbers, according to the peculiar genius of each language.
From some of the Australian tribes are derived
expressions still more complex, as for 6, marh-jin-bang-ga-gudjir-gyn,
half the hands and 1; and for 15, marh-jin-belli-belli-gudjir-jina-bang-ga,
the hand on either side and
half the feet.130 The Maré tribe, one of the numerous
island tribes of Melanesia,131 required for a translation
of the numeral 38, which occurs in John v. 5, “had an
infirmity thirty and eight years,” the circumlocution,
“one man and both sides five and three.” Such expressions,
curious as they seem at first thought, are no more
than the natural outgrowth of systems built up by the
slow and tedious process which so often obtains among
primitive races, where digit numerals are combined in an
almost endless variety of ways, and where mere reduplication
often serves in place of any independent names for
higher units. To what extent this may be carried is
shown by the language of the Cayubabi,132 who have for
10 the word tunca, and for 100 and 1000 the compounds
tunca tunca, and tunca tunca tunca respectively; or of the
Sapibocones, who call 10 bururuche, hand hand, and 100 buruche
buruche, hand hand hand hand.133 More remarkable
still is the Ojibwa language, which continues its numeral
scale without limit, furnishing combinations which are
really remarkable; as, e.g., that for 1,000,000,000, which
is me das wac me das wac as he me das wac,134 1000 × 1000
× 1000. The Winnebago expression for the same number,135
ho ke he hhuta hhu chen a ho ke he ka ra pa ne za
is no less formidable, but it has every appearance of
being an honest, native combination. All such primitive
terms for larger numbers must, however, be received
with caution. Savages are sometimes eager to display a
knowledge they do not possess, and have been known to
invent numeral words on the spot for the sake of carrying
their scales to as high a limit as possible. The
Choctaw words for million and billion are obvious attempts
to incorporate the corresponding English terms
into their own language.136 For million they gave the
vocabulary-hunter the phrase mil yan chuffa, and for billion,
bil yan chuffa. The word chuffa signifies 1, hence
these expressions are seen at a glance to be coined solely
for the purpose of gratifying a little harmless Choctaw
vanity. But this is innocence itself compared with the
fraud perpetrated on Labillardière by the Tonga Islanders,
who supplied the astonished and delighted investigator
with a numeral vocabulary up to quadrillions. Their
real limit was afterward found to be 100,000, and above
that point they had palmed off as numerals a tolerably
complete list of the obscene words of their language,
together with a few nonsense terms. These were all
accepted and printed in good faith, and the humiliating
truth was not discovered until years afterward.137
One noteworthy and interesting fact relating to
numeral nomenclature is the variation in form which
words of this class undergo when applied to different
classes of objects. To one accustomed as we are to
absolute and unvarying forms for numerals, this seems
at first a novel and almost unaccountable linguistic
freak. But it is not uncommon among uncivilized
races, and is extensively employed by so highly enlightened
a people, even, as the Japanese. This variation
in form is in no way analogous to that produced by
inflectional changes, such as occur in Hebrew, Greek,
Latin, etc. It is sufficient in many cases to produce
almost an entire change in the form of the word; or
to result in compounds which require close scrutiny for
the detection of the original root. For example, in
the Carrier, one of the Déné dialects of western Canada,
the word tha means 3 things; thane, 3 persons; that,
3 times; thatoen, in 3 places; thauh, in 3 ways; thailtoh,
all of the 3 things; thahoeltoh, all of the 3 persons;
and thahultoh, all of the 3 times.138 In the Tsimshian
language of British Columbia we find seven distinct
sets of numerals “which are used for various classes of
objects that are counted. The first set is used in
counting where there is no definite object referred to;
the second class is used for counting flat objects and
animals; the third for counting round objects and
divisions of time; the fourth for counting men; the
fifth for counting long objects, the numerals being composed
with kan, tree; the sixth for counting canoes;
and the seventh for measures. The last seem to be
composed with anon, hand.”139 The first ten numerals
of each of these classes is given in the following table:
No. | Counting | Flat Objects | Round Objects | Men | Long Objects | Canoes | Measures |
1gyakgakg'erelk'alk'awutskank'amaetk'al |
2t'epqatt'epqatgoupelt'epqadalgaopskang'alpēeltkgulbel |
3guantguantgutlegulalgaltskangaltskantkguleont |
4tqalpqtqalpqtqalpqtqalpqdaltqaapskantqalpqsktqalpqalont |
5kctōnckctōnckctōnckcenecalk'etoentskankctōonskkctonsilont |
6k'altk'altk'altk'aldalk'aoltskank'altkk'aldelont |
7t'epqaltt'epqaltt'epqaltt'epqaldalt'epqaltskant'epqaltkt'epqaldelont |
8guandaltyuktaltyuktaltyuktleadalek'tlaedskanyuktaltkyuktaldelont |
9kctemackctemackctemackctemacalkctemaestkankctemackkctemasilont |
10gy'apgy'apkpēelkpalkpēetskangy'apskkpeont |
Remarkable as this list may appear, it is by no
means as extensive as that derived from many of the
other British Columbian tribes. The numerals of the
Shushwap, Stlatlumh, Okanaken, and other languages
of this region exist in several different forms, and can
also be modified by any of the innumerable suffixes of
these tongues.140 To illustrate the almost illimitable
number of sets that may be formed, a table is given
of “a few classes, taken from the Heiltsuk dialect.141
It appears from these examples that the number of
classes is unlimited.”
| One. | Two. | Three. |
Animate. | menokmaalokyutuk
Round. | menskammasemyutqsem
Long. | ments'akmats'akyututs'ak
Flat. | menaqsamatlqsayutqsa
Day. | op'enequlsmatlp'enequlsyutqp'enequls
Fathom. | op'enkhmatlp'enkhyutqp'enkh
Grouped together. | ——matloutlyutoutl
Groups of objects. | nemtsmots'utlmatltsmots'utlyutqtsmots'utl
Filled cup. | menqtlalamatl'aqtlalayutqtlala
Empty cup. | menqtlamatl'aqtlayutqtla
Full box. | menskamalamasemalayutqsemala
Empty box. | menskammasemyutqsem
Loaded canoe. | mentsakemats'akeyututs'ake
Canoe with crew. | ments'akismats'aklayututs'akla
Together on beach. | ——maalis——
Together in house, etc. | ——maalitl——
Variation in numeral forms such as is exhibited in
the above tables is not confined to any one quarter of
the globe; but it is more universal among the British
Columbian Indians than among any other race, and it
is a more characteristic linguistic peculiarity of this
than of any other region, either in the Old World or
in the New. It was to some extent employed by the
Aztecs,142 and its use is current among the Japanese; in
whose language Crawfurd finds fourteen different classes
of numerals “without exhausting the list.”143
In examining the numerals of different languages it
will be found that the tens of any ordinary decimal
scale are formed in the same manner as in English.
Twenty is simply 2 times 10; 30 is 3 times 10, and
so on. The word “times” is, of course, not expressed,
any more than in English; but the expressions briefly
are, 2 tens, 3 tens, etc. But a singular exception to
this method is presented by the Hebrew, and other of
the Semitic languages. In Hebrew the word for 20
is the plural of the word for 10; and 30, 40, 50, etc.
to 90 are plurals of 3, 4, 5, 6, 7, 8, 9. These numerals
are as follows:144
10,eser,20,eserim, |
3,shalosh,30,shaloshim, |
4,arba,40,arbaim, |
5,chamesh,50,chamishshim, |
6,shesh,60,sheshshim, |
7,sheba,70,shibim, |
8,shemoneh,80,shemonim, |
9,tesha,90,tishim. |
The same formation appears in the numerals of the
ancient Phœnicians,
145 and seems, indeed, to be a well-marked
characteristic of the various branches of this
division of the Caucasian race. An analogous method
appears in the formation of the tens in the Bisayan,146
one of the Malay numeral scales, where 30, 40, …
90, are constructed from 3, 4, … 9, by adding the
termination -an.
No more interesting contribution has ever been made
to the literature of numeral nomenclature than that in
which Dr. Trumbull embodies the results of his scholarly
research among the languages of the native Indian
tribes of this country.147 As might be expected, we are
everywhere confronted with a digital origin, direct or
indirect, in the great body of the words examined.
But it is clearly shown that such a derivation cannot
be established for all numerals; and evidence collected
by the most recent research fully substantiates the position
taken by Dr. Trumbull. Nearly all the derivations
established are such as to remind us of the meanings
we have already seen recurring in one form or another
in language after language. Five is the end of the
finger count on one hand—as, the Micmac nan, and
Mohegan nunon, gone, or spent; the Pawnee sihuks,
hands half; the Dakota zaptan, hand turned down;
and the Massachusetts napanna, on one side. Ten is
the end of the finger count, but is not always expressed
by the “both hands” formula so commonly met with.
The Cree term for this number is mitatat, no further;
and the corresponding word in Delaware is m'tellen, no
more. The Dakota 10 is, like its 5, a straightening
out of the fingers which have been turned over in
counting, or wickchemna, spread out unbent. The same
is true of the Hidatsa pitika, which signifies a smoothing
out, or straightening. The Pawnee 4, skitiks, is
unusual, signifying as it does “all the fingers,” or more
properly, “the fingers of the hand.” The same meaning
attaches to this numeral in a few other languages
also, and reminds one of the habit some people have
of beginning to count on the forefinger and proceeding
from there to the little finger. Can this have been the
habit of the tribes in question? A suggestion of the
same nature is made by the Illinois and Miami words
for 8, parare and polane, which signify “nearly ended.”
Six is almost always digital in origin, though the derivation
may be indirect, as in the Illinois kakatchui,
passing beyond the middle; and the Dakota shakpe,
1 in addition. Some of these significations are well
matched by numerals from the Ewe scales of western
Africa, where we find the following:148
In studying the names for 2 we are at once led away
from a strictly digital origin for the terms by which
this number is expressed. These names seem to come
from four different sources: (1) roots denoting separation
or distinction; (2) likeness, equality, or opposition;
(3) addition,
i.e. putting to, or putting with; (4) coupling,
pairing, or matching. They are often related to,
and perhaps derived from, names of natural pairs, as
feet, hands, eyes, arms, or wings. In the Dakota and
Algonkin dialects 2 is almost always related to “arms”
or “hands,” and in the Athapaskan to “feet.” But the
relationship is that of common origin, rather than of
derivation from these pair-names. In the Puri and
Hottentot languages, 2 and “hand” are closely allied;
while in Sanskrit, 2 may be expressed by any one of
the words
kara, hand,
bahu, arm,
paksha, wing, or
netra,
eye.
149 Still more remote from anything digital in their
derivation are the following, taken at random from a
very great number of examples that might be cited to
illustrate this point. The Assiniboines call 7, shak ko
we, or u she nah, the odd number.150 The Crow 1, hamat,
signifies “the least”;151 the Mississaga 1, pecik, a very
small thing.152 In Javanese, Malay, and Manadu, the
words for 1, which are respectively siji, satu, and
sabuah, signify 1 seed, 1 pebble, and 1 fruit respectively153—words
as natural and as much to be expected at the
beginning of a number scale as any finger name could
possibly be. Among almost all savage races one form
or another of palpable arithmetic is found, such as
counting by seeds, pebbles, shells, notches, or knots;
and the derivation of number words from these sources
can constitute no ground for surprise. The Marquesan
word for 4 is pona, knot, from the practice of tying
breadfruit in knots of 4. The Maori 10 is tekau,
bunch, or parcel, from the counting of yams and fish
by parcels of 10.154 The Javanese call 25, lawe, a thread,
or string; 50, ekat, a skein of thread; 400, samas, a bit
of gold; 800, domas, 2 bits of gold.155 The Macassar
and Butong term for 100 is bilangan, 1 tale or reckoning.156
The Aztec 20 is cem pohualli, 1 count; 400 is
centzontli, 1 hair of the head; and 8000 is xiquipilli,
sack.157 This sack was of such a size as to contain 8000
cacao nibs, or grains, hence the derivation of the word
in its numeral sense is perfectly natural. In Japanese
we find a large number of terms which, as applied to
the different units of the number scale, seem almost
purely fanciful. These words, with their meanings as
given by a Japanese lexicon, are as follows:
10,000, or 104,män= enormous number.
108,oku= a compound of the words “man” and “mind.”
1012,chio= indication, or symptom.
1016,kei= capital city.
1020,si= a term referring to grains.
1024,owi= ——
1028,jio= extent of land.
1032,ko= canal.
1036,kan= some kind of a body of water.
1040,sai= justice.
1044,sā= support.
1048,kioku= limit, or more strictly, ultimate.
.012,rin= ——
.013,mo= hair (of some animal).
.014,shi= thread.
In addition to these, some of the lower fractional
values are described by words meaning “very small,”
“very fine thread,” “sand grain,” “dust,” and “very
vague.” Taken altogether, the Japanese number system
is the most remarkable I have ever examined, in the
extent and variety of the higher numerals with well-defined
descriptive names. Most of the terms employed
are such as to defy any attempt to trace the process
of reasoning which led to their adoption. It is not
improbable that the choice was, in some of these cases
at least, either accidental or arbitrary; but still, the
changes in word meanings which occur with the lapse
of time may have differentiated significations originally
alike, until no trace of kinship would appear to the
casual observer. Our numerals “score” and “gross”
are never thought of as having any original relation to
what is conveyed by the other meanings which attach
to these words. But the origin of each, which is easily
traced, shows that, in the beginning, there existed a
well-defined reason for the selection of these, rather
than other terms, for the numbers they now describe.
Possibly these remarkable Japanese terms may be
accounted for in the same way, though the supposition
is, for some reasons, quite improbable. The same may
be said for the Malagasy 1000, alina, which also means
“night,” and the Hebrew 6, shesh, which has the
additional signification “white marble,” and the stray
exceptions which now and then come to the light in
this or that language. Such terms as these may admit
of some logical explanation, but for the great mass of
numerals whose primitive meanings can be traced at
all, no explanation whatever is needed; the words are
self-explanatory, as the examples already cited show.
A few additional examples of natural derivation may
still further emphasize the point just discussed. In
Bambarese the word for 10, tank, is derived directly
from adang, to count.158 In the language of Mota, one of
the islands of Melanesia, 100 is mel nol, used and done
with, referring to the leaves of the cycas tree, with
which the count had been carried on.159 In many other
Melanesian dialects160 100 is rau, a branch or leaf. In
the Torres Straits we find the same number expressed
by na won, the close; and in Eromanga it is narolim
narolim (2 × 5)(2 × 5).161 This combination deserves
remark only because of the involved form which seems
to have been required for the expression of so small
a number as 100. A compound instead of a simple
term for any higher unit is never to be wondered at,
so rude are some of the savage methods of expressing
number; but “two fives (times) two fives” is certainly
remarkable. Some form like that employed by the Nusqually162
of Puget Sound for 1000, i.e. paduts-subquätche,
ten hundred, is more in accordance with primitive
method. But we are equally likely to find such descriptive
phrases for this numeral as the dor paka, banyan
roots, of the Torres Islands; rau na hai, leaves of a
tree, of Vaturana; or udolu, all, of the Fiji Islands.
And two curious phrases for 1000 are those of the
Banks' Islands, tar mataqelaqela, eye blind thousand, i.e.
many beyond count; and of Malanta, warehune huto,
opossum's hairs, or idumie one, count the sand.163
The native languages of India, Thibet, and portions
of the Indian archipelago furnish us with abundant
instances of the formation of secondary numeral scales,
which were used only for special purposes, and without
in any way interfering with the use of the number words
already in use. “Thus the scholars of India, ages ago,
selected a set of words for a memoria technica, in order
to record dates and numbers. These words they chose
for reasons which are still in great measure evident;
thus ‘moon’ or ‘earth’ expressed 1, there being but
one of each; 2 might be called ‘eye,’ ‘wing,’ ‘arm,’
‘jaw,’ as going in pairs; for 3 they said ‘Rama,’ ‘fire,’
or ‘quality,’ there being considered to be three Ramas,
three kinds of fire, three qualities (guna); for 4 were
used ‘veda,’ ‘age,’ or ‘ocean,’ there being four of each
recognized; ‘season’ for 6, because they reckoned six
seasons; ‘sage’ or ‘vowel,’ for 7, from the seven sages
and the seven vowels; and so on with higher numbers,
‘sun’ for 12, because of his twelve annual denominations,
or ‘zodiac’ from his twelve signs, and ‘nail’ for
20, a word incidentally bringing in finger notation. As
Sanskrit is very rich in synonyms, and as even the
numerals themselves might be used, it became very
easy to draw up phrases or nonsense verses to record
series of numbers by this system of artificial memory.”164
More than enough has been said to show how baseless
is the claim that all numeral words are derived, either
directly or indirectly, from the names of fingers, hands,
or feet. Connected with the origin of each number
word there may be some metaphor, which cannot always
be distinctly traced; and where the metaphor was born
of the hand or of the foot, we inevitably associate it
with the practice of finger counting. But races as fond
of metaphor and of linguistic embellishment as are those
of the East, or as are our American Indians even, might
readily resort to some other source than that furnished
by the members of the human body, when in want of
a term with which to describe the 5, 10, or any other
number of the numeral scale they were unconsciously
forming. That the first numbers of a numeral scale
are usually derived from other sources, we have some
reason to believe; but that all above 2, 3, or at most
4, are almost universally of digital origin we must admit.
Exception should properly be made of higher units, say
1000 or anything greater, which could not be expected
to conform to any law of derivation governing the first
few units of a system.
Collecting together and comparing with one another
the great mass of terms by which we find any number
expressed in different languages, and, while admitting
the great diversity of method practised by different
tribes, we observe certain resemblances which were not
at first supposed to exist. The various meanings of 1,
where they can be traced at all, cluster into a little
group of significations with which at last we come to
associate the idea of unity. Similarly of 2, or 5, or 10,
or any one of the little band which does picket duty
for the advance guard of the great host of number
words which are to follow. A careful examination of
the first decade warrants the assertion that the probable
meaning of any one of the units will be found in
the list given below. The words selected are intended
merely to serve as indications of the thought underlying
the savage's choice, and not necessarily as the
exact term by means of which he describes his number.
Only the commonest meanings are included in
the tabulation here given.
1= existence, piece, group, beginning.
2= repetition, division, natural pair.
3= collection, many, two-one.
4= two twos.
5= hand, group, division,
6= five-one, two threes, second one.
7= five-two, second two, three from ten.
8= five-three, second three, two fours, two from ten.
9= five-four, three threes, one from ten.
10= one (group), two fives (hands), half a man, one man.
15= ten-five, one foot, three fives.
20= two tens, one man, two feet.165
Miscellaneous Number Bases.
In the development and extension of any series of
numbers into a systematic arrangement to which the
term
system may be applied, the first and most indispensable
step is the selection of some number which is
to serve as a base. When the savage begins the process
of counting he invents, one after another, names
with which to designate the successive steps of his
numerical journey. At first there is no attempt at
definiteness in the description he gives of any considerable
number. If he cannot show what he means by
the use of his fingers, or perhaps by the fingers of a
single hand, he unhesitatingly passes it by, calling it
many, heap, innumerable, as many as the leaves on the
trees, or something else equally expressive and equally
indefinite. But the time comes at last when a greater
degree of exactness is required. Perhaps the number
11 is to be indicated, and indicated precisely. A fresh
mental effort is required of the ignorant child of
nature; and the result is “all the fingers and one
more,” “both hands and one more,” “one on another
count,” or some equivalent circumlocution. If he has
an independent word for 10, the result will be simply
ten-one. When this step has been taken, the base is
established. The savage has, with entire unconsciousness,
made all his subsequent progress dependent on
the number 10, or, in other words, he has established
10 as the base of his number system. The process just
indicated may be gone through with at 5, or at 20,
thus giving us a quinary or a vigesimal, or, more probably,
a mixed system; and, in rare instances, some
other number may serve as the point of departure
from simple into compound numeral terms. But the
general idea is always the same, and only the details
of formation are found to differ.
Without the establishment of some base any system
of numbers is impossible. The savage has no means of
keeping track of his count unless he can at each step
refer himself to some well-defined milestone in his
course. If, as has been pointed out in the foregoing
chapters, confusion results whenever an attempt is made
to count any number which carries him above 10, it
must at once appear that progress beyond that point
would be rendered many times more difficult if it were
not for the fact that, at each new step, he has only to
indicate the distance he has progressed beyond his base,
and not the distance from his original starting-point.
Some idea may, perhaps, be gained of the nature of
this difficulty by imagining the numbers of our ordinary
scale to be represented, each one by a single
symbol different from that used to denote any other
number. How long would it take the average intellect
to master the first 50 even, so that each number could
without hesitation be indicated by its appropriate symbol?
After the first 50 were once mastered, what of
the next 50? and the next? and the next? and so on.
The acquisition of a scale for which we had no other
means of expression than that just described would be
a matter of the extremest difficulty, and could never,
save in the most exceptional circumstances, progress
beyond the attainment of a limit of a few hundred.
If the various numbers in question were designated by
words instead of by symbols, the difficulty of the task
would be still further increased. Hence, the establishment
of some number as a base is not only a matter
of the very highest convenience, but of absolute necessity,
if any save the first few numbers are ever to
be used.
In the selection of a base,—of a number from which
he makes a fresh start, and to which he refers the
next steps in his count,—the savage simply follows
nature when he chooses 10, or perhaps 5 or 20. But
it is a matter of the greatest interest to find that other
numbers have, in exceptional cases, been used for this
purpose. Two centuries ago the distinguished philosopher
and mathematician, Leibnitz, proposed a binary
system of numeration. The only symbols needed in
such a system would be 0 and 1. The number which
is now symbolized by the figure 2 would be represented
by 10; while 3, 4, 5, 6, 7, 8, etc., would appear
in the binary notation as 11, 100, 101, 110, 111, 1000,
etc. The difficulty with such a system is that it rapidly
grows cumbersome, requiring the use of so many
figures for indicating any number. But Leibnitz found
in the representation of all numbers by means of the
two digits 0 and 1 a fitting symbolization of the creation
out of chaos, or nothing, of the entire universe by
the power of the Deity. In commemoration of this
invention a medal was struck bearing on the obverse
the words
Numero Deus impari gaudet,
and on the reverse,
Omnibus ex nihilo ducendis sufficit Unum.166
This curious system seems to have been regarded with
the greatest affection by its inventor, who used every
endeavour in his power to bring it to the notice of
scholars and to urge its claims. But it appears to have
been received with entire indifference, and to have
been regarded merely as a mathematical curiosity.
Unknown to Leibnitz, however, a binary method of
counting actually existed during that age; and it is
only at the present time that it is becoming extinct.
In Australia, the continent that is unique in its flora,
its fauna, and its general topography, we find also this
anomaly among methods of counting. The natives,
who are to be classed among the lowest and the least
intelligent of the aboriginal races of the world, have
number systems of the most rudimentary nature, and
evince a decided tendency to count by twos. This
peculiarity, which was to some extent shared by the
Tasmanians, the island tribes of the Torres Straits,
and other aboriginal races of that region, has by some
writers been regarded as peculiar to their part of the
world; as though a binary number system were not
to be found elsewhere. This attempt to make out of
the rude and unusual method of counting which obtained
among the Australians a racial characteristic is
hardly justified by fuller investigation. Binary number
systems, which are given in full on another page,
are found in South America. Some of the Dravidian
scales are binary;167 and the marked preference, not
infrequently observed among savage races, for counting
by pairs, is in itself a sufficient refutation of this
theory. Still it is an unquestionable fact that this
binary tendency is more pronounced among the Australians
than among any other extensive number of
kindred races. They seldom count in words above
4, and almost never as high as 7. One of the most
careful observers among them expresses his doubt as
to a native's ability to discover the loss of two pins,
if he were first shown seven pins in a row, and then
two were removed without his knowledge.168 But he
believes that if a single pin were removed from the
seven, the Blackfellow would become conscious of its
loss. This is due to his habit of counting by pairs,
which enables him to discover whether any number
within reasonable limit is odd or even. Some of the
negro tribes of Africa, and of the Indian tribes of
America, have the same habit. Progression by pairs
may seem to some tribes as natural as progression by
single units. It certainly is not at all rare; and in
Australia its influence on spoken number systems is
most apparent.
Any number system which passes the limit 10 is
reasonably sure to have either a quinary, a decimal, or
a vigesimal structure. A binary scale could, as it is
developed in primitive languages, hardly extend to 20,
or even to 10, without becoming exceedingly cumbersome.
A binary scale inevitably suggests a wretchedly
low degree of mental development, which stands in the
way of the formation of any number scale worthy to be
dignified by the name of system. Take, for example,
one of the dialects found among the western tribes of
the Torres Straits, where, in general, but two numerals
are found to exist. In this dialect the method of counting
is:169
1.urapun. |
2.okosa. |
3.okosa urapun= 2-1.
4.okosa okosa= 2-2.
5.okosa okosa urapun= 2-2-1.
6.okosa okosa okosa= 2-2-2.
Anything above 6 they call ras, a lot.
For the sake of uniformity we may speak of this as
a “system.” But in so doing, we give to the legitimate
meaning of the word a severe strain. The customs and
modes of life of these people are not such as to require
the use of any save the scanty list of numbers given
above; and their mental poverty prompts them to call 3,
the first number above a single pair, 2-1. In the same
way, 4 and 6 are respectively 2 pairs and 3 pairs, while
5 is 1 more than 2 pairs. Five objects, however, they
sometimes denote by urapuni-getal, 1 hand. A precisely
similar condition is found to prevail respecting the arithmetic
of all the Australian tribes. In some cases only
two numerals are found, and in others three. But in
a very great number of the native languages of that
continent the count proceeds by pairs, if indeed it proceeds
at all. Hence we at once reject the theory that
Australian arithmetic, or Australian counting, is essentially
peculiar. It is simply a legitimate result, such
as might be looked for in any part of the world, of the
barbarism in which the races of that quarter of the world
were sunk, and in which they were content to live.
The following examples of Australian and Tasmanian
number systems show how scanty was the numerical
ability possessed by these tribes, and illustrate fully
their tendency to count by twos or pairs.
Murray River.170
1.enea. |
2.petcheval. |
3.petchevalenea= 2-1.
4.petcheval peteheval= 2-2.
Maroura.
1.nukee. |
2.barkolo. |
3.barkolo nuke= 2-1.
4.barkolo barkolo= 2-2.
Mort Noular.
1.gamboden. |
2.bengeroo. |
3.bengeroganmel= 2-1.
4.bengeroovor bengeroo= 2 + 2.
Wimmera.
1.keyap. |
2.pollit. |
3.pollit keyap= 2-1.
4.pollit pollit= 2-2.
Popham Bay.
1.motu. |
2.lawitbari. |
3.lawitbari-motu= 2-1.
Kamilaroi.171
1.mal. |
2.bularr. |
3.guliba. |
4.bularrbularr= 2-2.
5.bulaguliba= 2-3.
6.gulibaguliba= 3-3.
Port Essington.172
1.erad. |
2.nargarik. |
3.nargarikelerad= 2-1.
4.nargariknargarik= 2-2.
Warrego.
1.tarlina. |
2.barkalo. |
3.tarlina barkalo= 1-2.
Crocker Island.
1.roka. |
2.orialk. |
3.orialkeraroka= 2-1.
Warrior Island.173
1.woorapoo. |
2.ocasara. |
3.ocasara woorapoo= 2-1.
4.ocasara ocasara= 2-2.
Dippil.174
1.kalim. |
2.buller. |
3.boppa. |
4.buller gira buller= 2 + 2.
5.buller gira buller kalim= 2 + 2 + 1.
Moreton's Bay.176
1.kunner. |
2.budela. |
3.muddan. |
4.budela berdelu= 2-2.
Encounter Bay.177
1.yamalaitye. |
2.ningenk. |
3.nepaldar. |
4.kuko kuko= 2-2, or pair pair.
5.kuko kuko ki= 2-2-1.
6.kuko kuko kuko= 2-2-2.
7.kuko kuko kuko ki= 2-2-2-1.
Adelaide.178
1.kuma. |
2.purlaitye, or bula. |
3.marnkutye. |
4.yera-bula= pair 2.
5.yera-bula kuma= pair 2-1.
6.yera-bula purlaitye= pair 2.2.
Wiraduroi.179
1.numbai. |
2.bula. |
3.bula-numbai= 2-1.
4.bungu= many.
5.bungu-galan= very many.
Wirri-Wirri.180
1.mooray. |
2.boollar. |
3.belar mooray= 2-1.
4.boollar boollar= 2-2.
5.mongoonballa. |
6.mongun mongun. |
Cooper's Creek.181
1.goona. |
2.barkoola. |
3.barkoola goona= 2-1.
4.barkoola barkoola= 2-2.
Bourke, Darling River.182
1.neecha. |
2.boolla. |
4.boolla neecha= 2-1.
3.boolla boolla= 2-2.
Yit-tha.184
1.mo. |
2.thral. |
3.thral mo= 2-1.
4.thral thral= 2-2.
Port Darwin.185
1.kulagook. |
2.kalletillick. |
3.kalletillick kulagook= 2-1.
4.kalletillick kalletillick= 2-2.
Champion Bay.186
1.kootea. |
2.woothera. |
3.woothera kootea= 2-1.
4.woothera woothera= 2-2.
Belyando River.187
1.wogin. |
2.booleroo. |
3.booleroo wogin= 2-1.
4.booleroo booleroo= 2-2.
Warrego River.
1.onkera. |
2.paulludy. |
3.paulludy onkera= 2-1.
4.paulludy paulludy= 2-2.
Richmond River.
1.yabra. |
2.booroora. |
3.booroora yabra= 2-1.
4.booroora booroora= 2-2.
Port Macquarie.
1.warcol. |
2.blarvo. |
3.blarvo warcol= 2-1.
4.blarvo blarvo= 2-2.
Hill End.
1.miko. |
2.bullagut. |
3.bullagut miko= 2-1.
4.bullagut bullagut= 2-2.
Moneroo
1.boor. |
2.wajala, blala. |
3.blala boor= 2-1.
4.wajala wajala. |
Gonn Station.
1.karp. |
2.pellige. |
3.pellige karp= 2-1.
4.pellige pellige= 2-2.
Upper Yarra.
1.kaambo. |
2.benjero. |
3.benjero kaambo= 2-2.
4.benjero on benjero= 2-2.
Omeo.
1.bore. |
2.warkolala. |
3.warkolala bore= 2-1.
4.warkolala warkolala= 2-2.
Snowy River.
1.kootook. |
2.boolong. |
3.booloom catha kootook= 2 + 1.
4.booloom catha booloom= 2 + 2.
Ngarrimowro.
1.warrangen. |
2.platir. |
3.platir warrangen= 2-1.
4.platir platir= 2-2.
This Australian list might be greatly extended, but
the scales selected may be taken as representative
examples of Australian binary scales. Nearly all of
them show a structure too clearly marked to require
comment. In a few cases, however, the systems are
to be regarded rather as showing a trace of binary
structure, than as perfect examples of counting by
twos. Examples of this nature are especially numerous
in Curr's extensive list—the most complete collection
of Australian vocabularies ever made.
A few binary scales have been found in South
America, but they show no important variation on the
Australian systems cited above. The only ones I have
been able to collect are the following:
Bakairi.188
1.tokalole. |
2.asage. |
3.asage tokalo= 2-1.
4.asage asage= 2-2.
Zapara.189
1.nuquaqui. |
2.namisciniqui. |
3.haimuckumarachi. |
4.namisciniqui ckara maitacka= 2 + 2.
5.namisciniqui ckara maitacka nuquaqui= 2 pairs + 1.
6.haimuckumaracki ckaramsitacka= 3 pairs.
Apinages.190
1.pouchi. |
2.at croudou. |
3.at croudi-pshi= 2-1.
4.agontad-acroudo= 2-2.
Cotoxo.191
1.ihueto. |
2.ize. |
3.ize-te-hueto= 2-1.
4.ize-te-seze= 2-2.
5.ize-te-seze-hue= 2-2-1.
Mbayi.192
1.uninitegui. |
2.iniguata. |
3.iniguata dugani= 2 over.
4.iniguata driniguata= 2-2.
5.oguidi= many.
Tama.193
1.teyo. |
2.cayapa. |
3.cho-teyo= 2 + 1.
4.cayapa-ria= 2 again.
5.cia-jente= hand.
Curetu.194
1.tchudyu. |
2.ap-adyu. |
3.arayu. |
4.apaedyái= 2 + 2.
5.tchumupa. |
If the existence of number systems like the above are
to be accounted for simply on the ground of low civilization,
one might reasonably expect to find ternary and
and quaternary scales, as well as binary. Such scales
actually exist, though not in such numbers as the binary.
An example of the former is the Betoya scale,195 which
runs thus:
1.edoyoyoi. |
2.edoi= another.
3.ibutu= beyond.
4.ibutu-edoyoyoi= beyond 1, or 3-1.
5.ru-mocoso= hand.
The Kamilaroi scale, given as an example of binary
formation, is partly ternary; and its word for 6,
guliba
guliba, 3-3, is purely ternary. An occasional ternary
trace is also found in number systems otherwise decimal
or quinary vigesimal; as the
dlkunoutl, second 3, of the
Haida Indians of British Columbia. The Karens of
India
196 in a system otherwise strictly decimal, exhibit
the following binary-ternary-quaternary vagary:
6.then tho= 3 × 2.
7.then tho ta= 3 × 2-1.
8.lwie tho= 4 × 2.
9.lwie tho ta= 4 × 2-1.
In the Wokka dialect,197 found on the Burnett River,
Australia, a single ternary numeral is found, thus:
1.karboon. |
2.wombura. |
3.chrommunda. |
4.chrommuda karboon= 3-1.
Instances of quaternary numeration are less rare than
are those of ternary, and there is reason to believe that
this method of counting has been practised more extensively
than any other, except the binary and the three
natural methods, the quinary, the decimal, and the
vigesimal. The number of fingers on one hand is,
excluding the thumb, four. Possibly there have been
tribes among which counting by fours arose as a legitimate,
though unusual, result of finger counting; just
as there are, now and then, individuals who count on
their fingers with the forefinger as a starting-point.
But no such practice has ever been observed among
savages, and such theorizing is the merest guess-work.
Still a definite tendency to count by fours is sometimes
met with, whatever be its origin. Quaternary traces
are repeatedly to be found among the Indian languages
of British Columbia. In describing the Columbians,
Bancroft says: “Systems of numeration are simple, proceeding
by fours, fives, or tens, according to the different
languages.…”198 The same preference for four is said
to have existed in primitive times in the languages of
Central Asia, and that this form of numeration, resulting
in scores of 16 and 64, was a development of finger
counting.199
In the Hawaiian and a few other languages of the
islands of the central Pacific, where in general the number
systems employed are decimal, we find a most interesting
case of the development, within number scales
already well established, of both binary and quaternary
systems. Their origin seems to have been perfectly
natural, but the systems themselves must have been
perfected very slowly. In Tahitian, Rarotongan, Mangarevan,
and other dialects found in the neighbouring
islands of those southern latitudes, certain of the higher
units, tekau, rau, mano, which originally signified 10, 100,
1000, have become doubled in value, and now stand for
20, 200, 2000. In Hawaiian and other dialects they have
again been doubled, and there they stand for 40, 400,
4000.200 In the Marquesas group both forms are found,
the former in the southern, the latter in the northern,
part of the archipelago; and it seems probable that one
or both of these methods of numeration are scattered
somewhat widely throughout that region. The origin
of these methods is probably to be found in the fact
that, after the migration from the west toward the east,
nearly all the objects the natives would ever count in
any great numbers were small,—as yams, cocoanuts,
fish, etc.,—and would be most conveniently counted
by pairs. Hence the native, as he counted one pair,
two pairs, etc., might readily say one, two, and so on,
omitting the word “pair” altogether. Having much more
frequent occasion to employ this secondary than the
primary meaning of his numerals, the native would easily
allow the original significations to fall into disuse, and
in the lapse of time to be entirely forgotten. With a
subsequent migration to the northward a second duplication
might take place, and so produce the singular
effect of giving to the same numeral word three different
meanings in different parts of Oceania. To illustrate
the former or binary method of numeration, the Tahuatan,
one of the southern dialects of the Marquesas group,
may be employed.201 Here the ordinary numerals are:
1.tahi,.
10.onohuu.
20.takau.
200.au.
2,000.mano.
20,000.tini.
200,000.tufa.
2,000,000.pohi.
In counting fish, and all kinds of fruit, except breadfruit,
the scale begins with
tauna, pair, and then,
omitting
onohuu, they employ the same words again,
but in a modified sense.
Takau becomes 10,
au 100,
etc.; but as the word “pair” is understood in each case,
the value is the same as before. The table formed on
this basis would be:
2 (units)= 1 tauna= 2. |
10 tauna= 1 takau= 20.
10 takau= 1 au= 200.
10 au= 1 mano= 2000.
10 mano= 1 tini= 20,000.
10 tini= 1 tufa= 200,000.
10 tufa= 1 pohi= 2,000,000.
For counting breadfruit they use
pona, knot, as their
unit, breadfruit usually being tied up in knots of
four.
Takau now takes its third signification, 40, and
becomes the base of their breadfruit system, so to
speak. For some unknown reason the next unit, 400,
is expressed by
tauau, while
au, which is the term that
would regularly stand for that number, has, by a second
duplication, come to signify 800. The next unit, mano,
has in a similar manner been twisted out of its original
sense, and in counting breadfruit is made to serve for
8000. In the northern, or Nukuhivan Islands, the
decimal-quaternary system is more regular. It is in
the counting of breadfruit only,202
4 breadfruits= 1 pona= 4. |
10 pona= 1 toha= 40.
10 toha= 1 au= 400.
10 au= 1 mano= 4000.
10 mano= 1 tini= 40,000.
10 tini= 1 tufa= 400,000.
10 tufa= 1 pohi= 4,000,000.
In the Hawaiian dialect this scale is, with slight
modification, the universal scale, used not only in
counting breadfruit, but any other objects as well.
The result is a complete decimal-quaternary system,
such as is found nowhere else in the world except in
this and a few of the neighbouring dialects of the
Pacific. This scale, which is almost identical with the
Nukuhivan, is
203
4 units= 1 ha or tauna= 4. |
10 tauna= 1 tanaha= 40.
10 tanaha= 1 lau= 400.
10 lau= 1 mano= 4000.
10 mano= 1 tini= 40,000.
10 tini= 1 lehu= 400,000.
The quaternary element thus introduced has modified
the entire structure of the Hawaiian number system.
Fifty is tanaha me ta umi, 40 + 10; 76 is 40 + 20 + 10
+ 6; 100 is ua tanaha ma tekau, 2 × 40 + 10; 200 is
lima tanaha, 5 × 40; and 864,895 is 2 × 400,000 + 40,000 +
6 × 4000 + 2 × 400 + 2 × 40 + 10 + 5.204 Such examples show
that this secondary influence, entering and incorporating
itself as a part of a well-developed decimal system,
has radically changed it by the establishment of 4 as
the primary number base. The role which 10 now
plays is peculiar. In the natural formation of a
quaternary scale new units would be introduced at 16,
64, 256, etc.; that is, at the square, the cube, and each
successive power of the base. But, instead of this, the
new units are introduced at 10 × 4, 100 × 4, 1000 × 4,
etc.; that is, at the products of 4 by each successive
power of the old base. This leaves the scale a decimal
scale still, even while it may justly be called quaternary;
and produces one of the most singular and interesting
instances of number-system formation that has
ever been observed. In this connection it is worth
noting that these Pacific island number scales have
been developed to very high limits—in some cases into
the millions. The numerals for these large numbers
do not seem in any way indefinite, but rather to convey
to the mind of the native an idea as clear as can well
be conveyed by numbers of such magnitude. Beyond
the limits given, the islanders have indefinite expressions,
but as far as can be ascertained these are only used
when the limits given above have actually been passed.
To quote one more example, the Hervey Islanders, who
have a binary-decimal scale, count as follows:
5 kaviri (bunches of cocoanuts)= 1 takau= 20. |
10 takau= 1 rau= 200.
10 rau= 1 mano= 2000.
10 mano= 1 kiu= 20,000.
10 kiu= 1 tini= 200,000.
Anything above this they speak of in an uncertain
way, as
mano mano or
tini tini, which may, perhaps,
be paralleled by our English phrases “myriads upon
myriads,” and “millions of millions.”
205 It is most remarkable
that the same quarter of the globe should
present us with the stunted number sense of the
Australians, and, side by side with it, so extended and
intelligent an appreciation of numerical values as that
possessed by many of the lesser tribes of Polynesia.
The Luli of Paraguay206 show a decided preference
for the base 4. This preference gives way only when
they reach the number 10, which is an ordinary digit
numeral. All numbers above that point belong rather
to decimal than to quaternary numeration. Their numerals
are:
1.alapea. |
2.tamop. |
3.tamlip. |
4.lokep. |
5.lokep moile alapea= 4 with 1,
or is-alapea= hand 1. |
6.lokep moile tamop= 4 with 2.
7.lokep moile tamlip= 4 with 3.
8.lokep moile lokep= 4 with 4.
9.lokep moile lokep alapea= 4 with 4-1.
10.is yaoum= all the fingers of hand.
11.is yaoum moile alapea= all the fingers of hand with 1.
20.is elu yaoum= all the fingers of hand and foot.
30.is elu yaoum moile is-yaoum = all the fingers of hand and foot with all the fingers of hand. |
Still another instance of quaternary counting, this
time carrying with it a suggestion of binary influence,
is furnished by the Mocobi
207 of the Parana region.
Their scale is exceedingly rude, and they use the fingers
and toes almost exclusively in counting; only
using their spoken numerals when, for any reason, they
wish to dispense with the aid of their hands and feet.
Their first eight numerals are:
1.iniateda. |
2.inabaca. |
3.inabacao caini= 2 above.
4.inabacao cainiba= 2 above 2;
or natolatata. |
5.inibacao cainiba iniateda= 2 above 2-1;
or natolatata iniateda= 4-1. |
6.natolatatata inibaca= 4-2.
7.natolata inibacao-caini= 4-2 above.
8.natolata-natolata= 4-4.
There is probably no recorded instance of a number
system formed on 6, 7, 8, or 9 as a base. No natural
reason exists for the choice of any of these numbers
for such a purpose; and it is hardly conceivable that
any race should proceed beyond the unintelligent
binary or quaternary stage, and then begin the formation
of a scale for counting with any other base than
one of the three natural bases to which allusion has
already been made. Now and then some anomalous
fragment is found imbedded in an otherwise regular
system, which carries us back to the time when the
savage was groping his way onward in his attempt to
give expression to some number greater than any he
had ever used before; and now and then one of these
fragments is such as to lead us to the border land of
the might-have-been, and to cause us to speculate on
the possibility of so great a numerical curiosity as a
senary or a septenary scale. The Bretons call 18 triouec'h,
3-6, but otherwise their language contains no hint
of counting by sixes; and we are left at perfect liberty
to theorize at will on the existence of so unusual
a number word. Pott remarks208 that the Bolans, of
western Africa, appear to make some use of 6 as their
number base, but their system, taken as a whole, is
really a quinary-decimal. The language of the Sundas,209
or mountaineers of Java, contains traces of senary counting.
The Akra words for 7 and 8, paggu and paniu,
appear to mean 6-1 and 7-1, respectively; and the same
is true of the corresponding Tambi words pagu and
panjo.210 The Watji tribe211 call 6 andee, and 7 anderee,
which probably means 6-1. These words are to be
regarded as accidental variations on the ordinary laws
of formation, and are no more significant of a desire
to count by sixes than is the Wallachian term deu-maw,
which expresses 18 as 2-9, indicates the existence of a
scale of which 9 is the base. One remarkably interesting
number system is that exhibited by the Mosquito
tribe212 of Central America, who possess an extensive
quinary-vigesimal scale containing one binary and
three senary compounds. The first ten words of this
singular scale, which has already been quoted, are:
1.kumi. |
2.wal.
3.niupa. |
4.wal-wal= 2-2.
5.mata-sip= fingers of one hand.
6.matlalkabe. |
7.matlalkabe pura kumi= 6 + 1.
8.matlalkabe pura wal= 6 + 2.
9.matlalkabe pura niupa= 6 + 3.
10.mata-wal-sip= fingers of the second hand.
In passing from 6 to 7, this tribe, also, has varied the
almost universal law of progression, and has called 7
6-1. Their 8 and 9 are formed in a similar manner;
but at 10 the ordinary method is resumed, and is continued
from that point onward. Few number systems
contain as many as three numerals which are associated
with 6 as their base. In nearly all instances we find
such numerals singly, or at most in pairs; and in the
structure of any system as a whole, they are of no importance
whatever. For example, in the Pawnee, a pure
decimal scale, we find the following odd sequence:213
6.shekshabish. |
7.petkoshekshabish= 2-6, i.e. 2d 6.
8.touwetshabish= 3-6, i.e. 3d 6.
9.loksherewa= 10 − 1.
In the Uainuma scale the expressions for 7 and 8 are
obviously referred to 6, though the meaning of 7 is
not given, and it is impossible to guess what it really
does signify. The numerals in question are:
214
6.aira-ettagapi. |
7.aira-ettagapi-hairiwigani-apecapecapsi. |
8.aira-ettagapi-matschahma= 6 + 2.
In the dialect of the Mille tribe a single trace of
senary counting appears, as the numerals given below
show:
215
6.dildjidji. |
7.dildjidji me djuun= 6 + 1.
Finally, in the numerals used by the natives of the
Marshall Islands, the following curiously irregular sequence
also contains a single senary numeral:
216
6.thil thino= 3 + 3.
7.thilthilim-thuon= 6 + 1.
8.rua-li-dok= 10 − 2.
9.ruathim-thuon= 10 − 2 + 1.
Many years ago a statement appeared which at once
attracted attention and awakened curiosity. It was to
the effect that the Maoris, the aboriginal inhabitants of
New Zealand, used as the basis of their numeral system
the number 11; and that the system was quite
extensively developed, having simple words for 121
and 1331, i.e. for the square and cube of 11. No apparent
reason existed for this anomaly, and the Maori
scale was for a long time looked upon as something
quite exceptional and outside all ordinary rules of
number-system formation. But a closer and more accurate
knowledge of the Maori language and customs
served to correct the mistake, and to show that this
system was a simple decimal system, and that the error
arose from the following habit. Sometimes when counting
a number of objects the Maoris would put aside 1
to represent each 10, and then those so set aside would
afterward be counted to ascertain the number of tens
in the heap. Early observers among this people, seeing
them count 10 and then set aside 1, at the same time
pronouncing the word tekau, imagined that this word
meant 11, and that the ignorant savage was making
use of this number as his base. This misconception
found its way into the early New Zealand dictionary,
but was corrected in later editions. It is here mentioned
only because of the wide diffusion of the error,
and the interest it has always excited.217
Aside from our common decimal scale, there exist in
the English language other methods of counting, some
of them formal enough to be dignified by the term
system—as the sexagesimal method of measuring time
and angular magnitude; and the duodecimal system of
reckoning, so extensively used in buying and selling.
Of these systems, other than decimal, two are noticed
by Tylor,218 and commented on at some length, as
follows:
“One is the well-known dicing set, ace, deuce, tray,
cater, cinque, size; thus size-ace is 6-1, cinques or sinks,
double 5. These came to us from France, and correspond
with the common French numerals, except ace,
which is Latin as, a word of great philological interest,
meaning ‘one.’ The other borrowed set is to be found
in the Slang Dictionary. It appears that the English
street-folk have adopted as a means of secret communication
a set of Italian numerals from the organ-grinders
and image-sellers, or by other ways through which
Italian or Lingua Franca is brought into the low
neighbourhoods of London. In so doing they have performed
a philological operation not only curious but
instructive. By copying such expressions as due soldi,
tre soldi, as equivalent to ‘twopence,’ ‘threepence,’ the
word saltee became a recognized slang term for ‘penny’;
and pence are reckoned as follows:
oney saltee1d.uno soldo.
dooe saltee2d.due soldi.
tray saltee3d.tre soldi.
quarterer saltee4d.quattro soldi.
chinker saltee5d.cinque soldi.
say saltee6d.sei soldi.
say oney saltee, or setter saltee7d.sette soldi.
say dooe saltee, or otter saltee8d.otto soldi.
say tray saltee, or nobba saltee9d.nove soldi.
say quarterer saltee, or dacha saltee10d.dieci soldi.
say chinker saltee or dacha oney saltee11d.undici soldi.
oney beong1s.
a beong say saltee1s. 6d.
dooe beong say saltee, or madza caroon2s. 6d.(half-crown, mezza corona).
One of these series simply adopts Italian numerals
decimally. But the other, when it has reached 6,
having had enough of novelty, makes 7 by 6-1, and so
forth. It is for no abstract reason that 6 is thus made
the turning-point, but simply because the costermonger is
adding pence up to the silver sixpence, and then adding
pence again up to the shilling. Thus our duodecimal
coinage has led to the practice of counting by sixes, and
produced a philological curiosity, a real senary notation.”
In addition to the two methods of counting here
alluded to, another may be mentioned, which is equally
instructive as showing how readily any special method
of reckoning may be developed out of the needs arising
in connection with any special line of work. As
is well known, it is the custom in ocean, lake, and
river navigation to measure soundings by the fathom.
On the Mississippi River, where constant vigilance is
needed because of the rapid shifting of sand-bars, a
special sounding nomenclature has come into vogue,219
which the following terms will illustrate:
5ft.= five feet. |
6ft.= six feet. |
9ft.= nine feet. |
10-1/2ft.= a quarter less twain; i.e. a quarter of a fathom less than 2.
12ft.= mark twain.
13-1/2ft.= a quarter twain.
16-1/2ft.= a quarter less three.
18ft.= mark three.
19-1/2ft.= a quarter three.
24ft.= deep four.
As the soundings are taken, the readings are called
off in the manner indicated in the table; 10-1/2 feet
being “a quarter less twain,” 12 feet “mark twain,”
etc. Any sounding above “deep four” is reported as
“no bottom.” In the Atlantic and Gulf waters on the
coast of this country the same system prevails, only it
is extended to meet the requirements of the deeper
soundings there found, and instead of “six feet,” “mark
twain,” etc., we find the fuller expressions, “by the
mark one,” “by the mark two,” and so on, as far as
the depth requires. This example also suggests the
older and far more widely diffused method of reckoning
time at sea by bells; a system in which “one
bell,” “two bells,” “three bells,” etc., mark the passage
of time for the sailor as distinctly as the hands of the
clock could do it. Other examples of a similar nature
will readily suggest themselves to the mind.
Two possible number systems that have, for purely
theoretical reasons, attracted much attention, are the
octonary and the duodecimal systems. In favour of the
octonary system it is urged that 8 is an exact power
of 2; or in other words, a large number of repeated
halves can be taken with 8 as a starting-point, without
producing a fractional result. With 8 as a base we
should obtain by successive halvings, 4, 2, 1. A similar
process in our decimal scale gives 5, 2-1/2, 1-1/4. All
this is undeniably true, but, granting the argument up
to this point, one is then tempted to ask “What of
it?” A certain degree of simplicity would thereby be
introduced into the Theory of Numbers; but the only
persons sufficiently interested in this branch of mathematics
to appreciate the benefit thus obtained are
already trained mathematicians, who are concerned
rather with the pure science involved, than with reckoning
on any special base. A slightly increased simplicity
would appear in the work of stockbrokers, and
others who reckon extensively by quarters, eighths, and
sixteenths. But such men experience no difficulty whatever
in performing their mental computations in the
decimal system; and they acquire through constant
practice such quickness and accuracy of calculation,
that it is difficult to see how octonary reckoning would
materially assist them. Altogether, the reasons that
have in the past been adduced in favour of this form of
arithmetic seem trivial. There is no record of any
tribe that ever counted by eights, nor is there the
slightest likelihood that such a system could ever meet
with any general favour. It is said that the ancient
Saxons used the octonary system,220 but how, or for
what purposes, is not stated. It is not to be supposed
that this was the common system of counting, for it is
well known that the decimal scale was in use as far
back as the evidence of language will take us. But
the field of speculation into which one is led by the
octonary scale has proved most attractive to some, and
the conclusion has been soberly reached, that in the
history of the Aryan race the octonary was to be regarded
as the predecessor of the decimal scale. In
support of this theory no direct evidence is brought
forward, but certain verbal resemblances. Those ignes
fatuii of the philologist are made to perform the duty
of supporting an hypothesis which would never have
existed but for their own treacherous suggestions.
Here is one of the most attractive of them:
Between the Latin words novus, new, and novem, nine,
there exists a resemblance so close that it may well be
more than accidental. Nine is, then, the new number;
that is, the first number on a new count, of which 8
must originally have been the base. Pursuing this
thought by investigation into different languages, the
same resemblance is found there. Hence the theory is
strengthened by corroborative evidence. In language
after language the same resemblance is found, until it
seems impossible to doubt, that in prehistoric times, 9
was the new number—the beginning of a second tale.
The following table will show how widely spread is
this coincidence:
Sanskrit, navan= 9.nava= new.
Persian, nuh= 9.nau= new.
Greek, ἐννέα= 9.νέος= new.
Latin, novem= 9.novus= new.
German, neun= 9.neu= new.
Swedish, nio= 9.ny= new.
Dutch, negen= 9.nieuw= new.
Danish, ni= 9.ny= new.
Icelandic, nyr= 9.niu= new.
English, nine= 9.new= new.
French, neuf= 9.nouveau= new.
Spanish, nueve= 9.neuvo= new.
Italian, nove= 9.nuovo= new.
Portuguese, nove= 9.novo= new.
Irish, naoi= 9.nus= new.
Welsh, naw= 9.newydd= new.
Breton, nevez= 9.nuhue= new.221
This table might be extended still further, but the
above examples show how widely diffused throughout
the Aryan languages is this resemblance. The list
certainly is an impressive one, and the student is at
first thought tempted to ask whether all these resemblances
can possibly have been accidental. But a single
consideration sweeps away the entire argument as
though it were a cobweb. All the languages through
which this verbal likeness runs are derived directly
or indirectly from one common stock; and the common
every-day words, “nine” and “new,” have been transmitted
from that primitive tongue into all these linguistic
offspring with but little change. Not only are
the two words in question akin in each individual language,
but
they are akin in all the languages. Hence
all these resemblances reduce to a single resemblance,
or perhaps identity, that between the Aryan words for
“nine” and “new.” This was probably an accidental
resemblance, no more significant than any one of the
scores of other similar cases occurring in every language.
If there were any further evidence of the
former existence of an Aryan octonary scale, the coincidence
would possess a certain degree of significance;
but not a shred has ever been produced which is
worthy of consideration. If our remote ancestors ever
counted by eights, we are entirely ignorant of the fact,
and must remain so until much more is known of their
language than scholars now have at their command.
The word resemblances noted above are hardly more
significant than those occurring in two Polynesian languages,
the Fatuhivan and the Nakuhivan,222 where
“new” is associated with the number 7. In the former
case 7 is fitu, and “new” is fou; in the latter 7 is
hitu, and “new” is hou. But no one has, because of this
likeness, ever suggested that these tribes ever counted
by the senary method. Another equally trivial resemblance
occurs in the Tawgy and the Kamassin languages,223
thus:
Tawgy.
8.siti-data= 2 × 4.
9.nameaitjuma= another.
Kamassin.
8.sin-the'de= 2 × 4.
9.amithun= another.
But it would be childish to argue, from this fact
alone, that either 4 or 8 was the number base used.
In a recent antiquarian work of considerable interest,
the author examines into the question of a former
octonary system of counting among the various races
of the world, particularly those of Asia, and brings to
light much curious and entertaining material respecting
the use of this number. Its use and importance in
China, India, and central Asia, as well as among some
of the islands of the Pacific, and in Central America,
leads him to the conclusion that there was a time, long
before the beginning of recorded history, when 8 was
the common number base of the world. But his conclusion
has no basis in his own material even. The
argument cannot be examined here, but any one who
cares to investigate it can find there an excellent illustration
of the fact that a pet theory may take complete
possession of its originator, and reduce him finally to a
state of infantile subjugation.224
Of all numbers upon which a system could be based,
12 seems to combine in itself the greatest number of
advantages. It is capable of division by 2, 3, 4, and 6,
and hence admits of the taking of halves, thirds, quarters,
and sixths of itself without the introduction of
fractions in the result. From a commercial stand-point
this advantage is very great; so great that many have
seriously advocated the entire abolition of the decimal
scale, and the substitution of the duodecimal in its
stead. It is said that Charles XII. of Sweden was
actually contemplating such a change in his dominions
at the time of his death. In pursuance of this idea,
some writers have gone so far as to suggest symbols
for 10 and 11, and to recast our entire numeral nomenclature
to conform to the duodecimal base.225 Were such
a change made, we should express the first nine numbers
as at present, 10 and 11 by new, single symbols,
and 12 by 10. From this point the progression would
be regular, as in the decimal scale—only the same
combination of figures in the different scales would
mean very different things. Thus, 17 in the decimal
scale would become 15 in the duodecimal; 144 in the
decimal would become 100 in the duodecimal; and
1728, the cube of the new base, would of course be
represented by the figures 1000.
It is impossible that any such change can ever meet
with general or even partial favour, so firmly has the
decimal scale become intrenched in its position. But it
is more than probable that a large part of the world of
trade and commerce will continue to buy and sell by the
dozen, the gross, or some multiple or fraction of the one
or the other, as long as buying and selling shall continue.
Such has been its custom for centuries, and such
will doubtless be its custom for centuries to come. The
duodecimal is not a natural scale in the same sense as
are the quinary, the decimal, and the vigesimal; but it
is a system which is called into being long after the
complete development of one of the natural systems,
solely because of the simple and familiar fractions into
which its base is divided. It is the scale of civilization,
just as the three common scales are the scales of nature.
But an example of its use was long sought for in vain
among the primitive races of the world. Humboldt, in
commenting on the number systems of the various peoples
he had visited during his travels, remarked that no race
had ever used exclusively that best of bases, 12. But
it has recently been announced226 that the discovery of
such a tribe had actually been made, and that the
Aphos of Benuë, an African tribe, count to 12 by
simple words, and then for 13 say 12-1, for 14, 12-2,
etc. This report has yet to be verified, but if true
it will constitute a most interesting addition to anthropological
knowledge.
The Quinary System.
The origin of the quinary mode of counting has been
discussed with some fulness in a preceding chapter,
and upon that question but little more need be said.
It is the first of the natural systems. When the savage
has finished his count of the fingers of a single
hand, he has reached this natural number base. At
this point he ceases to use simple numbers, and begins
the process of compounding. By some one of the
numerous methods illustrated in earlier chapters, he
passes from 5 to 10, using here the fingers of his
second hand. He now has two fives; and, just as we
say “twenty,” i.e. two tens, he says “two hands,”
“the second hand finished,” “all the fingers,” “the
fingers of both hands,” “all the fingers come to an
end,” or, much more rarely, “one man.” That is, he
is, in one of the many ways at his command, saying
“two fives.” At 15 he has “three hands” or “one
foot”; and at 20 he pauses with “four hands,” “hands
and feet,” “both feet,” “all the fingers of hands and
feet,” “hands and feet finished,” or, more probably,
“one man.” All these modes of expression are strictly
natural, and all have been found in the number scales
which were, and in many cases still are, in daily use
among the uncivilized races of mankind.
In its structure the quinary is the simplest, the most
primitive, of the natural systems. Its base is almost
always expressed by a word meaning “hand,” or by
some equivalent circumlocution, and its digital origin
is usually traced without difficulty. A consistent formation
would require the expression of 10 by some
phrase meaning “two fives,” 15 by “three fives,” etc.
Such a scale is the one obtained from the Betoya language,
already mentioned in Chapter III., where the formation
of the numerals is purely quinary, as the following
indicate:227
5.teente= 1 hand.
10.cayaente, or caya huena= 2 hands.
15.toazumba-ente= 3 hands.
20.caesa-ente= 4 hands.
The same formation appears, with greater or less distinctness,
in many of the quinary scales already quoted,
and in many more of which mention might be made.
Collecting the significant numerals from a few such
scales, and tabulating them for the sake of convenience
of comparison, we see this point clearly illustrated
by the following:
Tamanac.
5.amnaitone= 1 hand.
10.amna atse ponare= 2 hands.
Arawak, Guiana.
5.abba tekkabe= 1 hand.
10.biamantekkabe= 2 hands.
Jiviro.
5.alacötegladu= 1 hand.
10.catögladu= 2 hands.
Niam Niam
5.biswe |
10.bauwe= 2d 5.
Nengones
5.se dono= the end (of the fingers of 1 hand).
10.rewe tubenine= 2 series (of fingers).
Sesake.228
5.lima= hand.
10.dua lima= 2 hands.
Ambrym.229
5.lim= hand.
10.ra-lim= 2 hands.
Pama.229
5.e-lime= hand.
10.ha-lua-lim= the 2 hands.
Dinka.230
5.wdyets. |
10.wtyer, or wtyar= 5 × 2.
Bari
5.kanat |
10.puök= 5 + 5?
Kanuri
5.ugu. |
10.megu= 2 × 5.
Rio Norte and San Antonio.231
5.juyopamauj. |
10.juyopamauj ajte= 5 × 2.
Api.232
5.lima. |
10.lua-lima= 2 × 5.
Erromango
5.suku-rim. |
10.nduru-lim= 2 × 5.
Tlingit, British Columbia.233
5.kedjin (from djin = hand). |
10.djinkat= both hands?
Thus far the quinary formation is simple and regular;
and in view of the evidence with which these and
similar illustrations furnish us, it is most surprising to
find an eminent authority making the unequivocal statement
that the number 10 is nowhere expressed by 2
fives234—that all tribes which begin their count on a
quinary base express 10 by a simple word. It is a
fact, as will be fully illustrated in the following pages,
that quinary number systems, when extended, usually
merge into either the decimal or the vigesimal. The
result is, of course, a compound of two, and sometimes
of three, systems in one scale. A pure quinary or
vigesimal number system is exceedingly rare; but quinary
scales certainly do exist in which, as far as we
possess the numerals, no trace of any other influence
appears. It is also to be noticed that some tribes, like
the Eskimos of Point Barrow, though their systems may
properly be classed as mixed systems, exhibit a decided
preference for 5 as a base, and in counting objects, divided
into groups of 5, obtaining the sum in this way.235
But the savage, after counting up to 10, often finds
himself unconsciously impelled to depart from his strict
reckoning by fives, and to assume a new basis of reference.
Take, for example, the Zuñi system, in which
the first 2 fives are:
5.öpte= the notched off.
10.astem'thla= all the fingers.
It will be noticed that the Zuñi does not say “two
hands,” or “the fingers of both hands,” but simply “all
the fingers.” The 5 is no longer prominent, but instead
the mere notion of one entire count of the fingers has
taken its place. The division of the fingers into two sets
of five each is still in his mind, but it is no longer the
leading idea. As the count proceeds further, the quinary
base may be retained, or it may be supplanted by a decimal
or a vigesimal base. How readily the one or the
other may predominate is seen by a glance at the following
numerals:
Galibi.236
5.atoneigne oietonaï= 1 hand.
10.oia batoue= the other hand.
20.poupoupatoret oupoume= feet and hands.
40.opoupoume= twice the feet and hands.
Guarani.237
5.ace popetei= 1 hand.
10.ace pomocoi= 2 hands.
20.acepo acepiabe= hands and feet.
Fate.238
5.lima= hand.
10.relima= 2 hands.
20.relima rua= (2 × 5) × 2.
Kiriri
5.mibika misa= 1 hand.
10.mikriba misa sai= both hands.
20.mikriba nusa ideko ibi sai= both hands together with the feet.
Zamuco
5.tsuena yimana-ite= ended 1 hand.
10.tsuena yimana-die= ended both hands.
20.tsuena yiri-die= ended both feet.
Yaruros.239
5.kani-iktsi-mo= 1 hand alone.
10.yowa-iktsi-bo= all the hands.
15.kani-tao-mo= 1 foot alone.
20.kani-pume= 1 man.
By the time 20 is reached the savage has probably
allowed his conception of any aggregate to be so far
modified that this number does not present itself to
his mind as 4 fives. It may find expression in some
phraseology such as the Kiriris employ—“both hands
together with the feet”—or in the shorter “ended both
feet” of the Zamucos, in which case we may presume
that he is conscious that his count has been completed
by means of the four sets of fives which are furnished
by his hands and feet. But it is at least equally probable
that he instinctively divides his total into 2 tens,
and thus passes unconsciously from the quinary into the
decimal scale. Again, the summing up of the 10 fingers
and 10 toes often results in the concept of a single
whole, a lump sum, so to speak, and the savage then
says “one man,” or something that gives utterance to
this thought of a new unit. This leads the quinary into
the vigesimal scale, and produces the combination so
often found in certain parts of the world. Thus the
inevitable tendency of any number system of quinary
origin is toward the establishment of another and larger
base, and the formation of a number system in which
both are used. Wherever this is done, the greater of
the two bases is always to be regarded as the principal
number base of the language, and the 5 as entirely subordinate
to it. It is hardly correct to say that, as a
number system is extended, the quinary element disappears
and gives place to the decimal or vigesimal,
but rather that it becomes a factor of quite secondary
importance in the development of the scale. If, for
example, 8 is expressed by 5-3 in a quinary decimal
system, 98 will be 9 × 10 + 5-3. The quinary element
does not disappear, but merely sinks into a relatively
unimportant position.
One of the purest examples of quinary numeration
is that furnished by the Betoya scale, already given in
full in Chapter III., and briefly mentioned at the beginning
of this chapter. In the simplicity and regularity
of its construction it is so noteworthy that it is worth
repeating, as the first of the long list of quinary
systems given in the following pages. No further
comment is needed on it than that already made in
connection with its digital significance. As far as
given by Dr. Brinton the scale is:
1.tey. |
2.cayapa. |
3.toazumba. |
4.cajezea= 2 with plural termination.
5.teente= hand.
6.teyente tey= 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.
A far more common method of progression is furnished
by languages which interrupt the quinary formation
at 10, and express that number by a single
word. Any scale in which this takes place can, from
this point onward, be quinary only in the subordinate
sense to which allusion has just been made. Examples
of this are furnished in a more or less perfect manner
by nearly all so-called quinary-vigesimal and quinary-decimal
scales. As fairly representing this phase of
number-system structure, I have selected the first 20
numerals from the following languages:
Welsh.240
1.un. |
2.dau. |
3.tri. |
4.pedwar. |
5.pump. |
6.chwech. |
7.saith. |
8.wyth. |
9.naw. |
10.deg. |
11.un ar ddeg= 1 + 10.
12.deuddeg= 2 + 10.
13.tri ar ddeg= 3 + 10.
14.pedwar ar ddeg= 4 + 10.
15.pymtheg= 5 + 10.
16.un ar bymtheg= 1 + 5 + 10.
17.dau ar bymtheg= 2 + 5 + 10.
18.tri ar bymtheg= 3 + 5 + 10.
19.pedwar ar bymtheg= 4 + 5 + 10.
20.ugain. |
Nahuatl.241
1.ce. |
2.ome. |
3.yei. |
4.naui. |
5.macuilli. |
6.chiquacen= [5] + 1.
7.chicome= [5] + 2.
8.chicuey= [5] + 3.
9.chiucnaui= [5] + 4.
10.matlactli. |
11.matlactli oce= 10 + 1.
12.matlactli omome= 10 + 2.
13.matlactli omey= 10 + 3.
14.matlactli onnaui= 10 + 4.
15.caxtolli. |
16.caxtolli oce= 15 + 1.
17.caxtolli omome= 15 + 2.
18.caxtolli omey= 15 + 3.
19.caxtolli onnaui= 15 + 4.
20.cempualli= 1 account.
Canaque242 New Caledonia.
1.chaguin. |
2.carou. |
3.careri. |
4.caboue |
5.cani. |
6.cani-mon-chaguin= 5 + 1.
7.cani-mon-carou= 5 + 2.
8.cani-mon-careri= 5 + 3.
9.cani-mon-caboue= 5 + 4.
10.panrere. |
11.panrere-mon-chaguin= 10 + 1.
12.panrere-mon-carou= 10 + 2.
13.panrere-mon-careri= 10 + 3.
14.panrere-mon-caboue= 10 + 4.
15.panrere-mon-cani= 10 + 5.
16.panrere-mon-cani-mon-chaguin= 10 + 5 + 1.
17.panrere-mon-cani-mon-carou= 10 + 5 + 2.
18.panrere-mon-cani-mon-careri= 10 + 5 + 3.
19.panrere-mon-cani-mon-caboue= 10 + 5 + 4.
20.jaquemo= 1 person.
Guato.243
1.cenai. |
2.dououni. |
3.coum. |
4.dekai. |
5.quinoui. |
6.cenai-caicaira= 1 on the other?
7.dououni-caicaira= 2 on the other?
8.coum-caicaira= 3 on the other?
9.dekai-caicaira= 4 on the other?
10.quinoi-da= 5 × 2.
11.cenai-ai-caibo= 1 + (the) hands.
12.dououni-ai-caibo= 2 + 10.
13.coum-ai-caibo= 3 + 10.
14.dekai-ai-caibo= 4 + 10.
15.quin-oibo= 5 × 3.
16.cenai-ai-quacoibo= 1 + 15.
17.dououni-ai-quacoibo= 2 + 15.
18.coum-ai-quacoibo= 3 + 15.
19.dekai-ai-quacoibo= 4 + 15.
20.quinoui-ai-quacoibo= 5 + 15.
The meanings assigned to the numerals 6 to 9 are entirely
conjectural. They obviously mean 1, 2, 3, 4, taken
a second time, and as the meanings I have given are
often found in primitive systems, they have, at a venture,
been given here.
Lifu, Loyalty Islands.244
1.ca. |
2.lue. |
3.koeni. |
4.eke. |
5.tji pi. |
6.ca ngemen= 1 above.
7.lue ngemen= 2 above.
8.koeni ngemen= 3 above.
9.eke ngemen= 4 above.
10.lue pi= 2 × 5.
11.ca ko. |
12.lue ko. |
13.koeni ko. |
14.eke ko. |
15.koeni pi= 3 × 5.
16.ca huai ano. |
17.lua huai ano. |
18.koeni huai ano. |
19.eke huai ano. |
20.ca atj= 1 man.
Bongo.245
1.kotu. |
2.ngorr. |
3.motta. |
4.neheo. |
5.mui. |
6.dokotu= [5] + 1.
7.dongorr= [5] + 2.
8.domotta= [5] + 3.
9.doheo= [5] + 4.
10.kih. |
11.ki dokpo kotu= 10 + 1.
12.ki dokpo ngorr= 10 + 2.
13.ki dokpo motta= 10 + 3.
14.ki dokpo neheo= 10 + 4.
15.ki dokpo mui= 10 + 5.
16.ki dokpo mui do mui okpo kotu= 10 + 5 more, to 5, 1 more.
17.ki dokpo mui do mui okpo ngorr= 10 + 5 more, to 5, 2 more.
18.ki dokpo mui do mui okpo motta= 10 + 5 more, to 5, 3 more.
19.ki dokpo mui do mui okpo nehea= 10 + 5 more, to 5, 4 more.
20.mbaba kotu. |
Above 20, the Lufu and the Bongo systems are vigesimal,
so that they are, as a whole, mixed systems.
The Welsh scale begins as though it were to present
a pure decimal structure, and no hint of the quinary
element appears until it has passed 15. The Nahuatl,
on the other hand, counts from 5 to 10 by the ordinary
quinary method, and then appears to pass into the decimal
form. But when 16 is reached, we find the quinary
influence still persistent; and from this point to 20, the
numeral words in both scales are such as to show that
the notion of counting by fives is quite as prominent as
the notion of referring to 10 as a base. Above 20 the
systems become vigesimal, with a quinary or decimal
structure appearing in all numerals except multiples of
20. Thus, in Welsh, 36 is unarbymtheg ar ugain, 1 + 5
+ 10 + 20; and in Nahuatl the same number is cempualli
caxtolli oce, 20 + 15 + 1. Hence these and similar number
systems, though commonly alluded to as vigesimal,
are really mixed scales, with 20 as their primary base.
The Canaque scale differs from the Nahuatl only in
forming a compound word for 15, instead of introducing
a new and simple term.
In the examples which follow, it is not thought best
to extend the lists of numerals beyond 10, except in
special instances where the illustration of some particular
point may demand it. The usual quinary scale will be
found, with a few exceptions like those just instanced,
to have the following structure or one similar to it in all
essential details: 1, 2, 3, 4, 5, 5-1, 5-2, 5-3, 5-4, 10,
10-1, 10-2, 10-3, 10-4, 10-5, 10-5-1, 10-5-2, 10-5-3,
10-5-4, 20. From these forms the entire system can
readily be constructed as soon as it is known whether
its principal base is to be 10 or 20.
Turning first to the native African languages, I have
selected the following quinary scales from the abundant
material that has been collected by the various explorers
of the “Dark Continent.” In some cases the numerals
of certain tribes, as given by one writer, are found to
differ widely from the same numerals as reported by
another. No attempt has been made at comparison of
these varying forms of orthography, which are usually
to be ascribed to difference of nationality on the part
of the collectors.
Feloops.246
1.enory. |
2.sickaba, or cookaba. |
3.sisajee. |
4.sibakeer. |
5.footuck. |
6.footuck-enory= 5-1.
7.footuck-cookaba= 5-2.
8.footuck-sisajee= 5-3.
9.footuck-sibakeer= 5-4.
10.sibankonyen. |
Kissi.247
1.pili. |
2.miu. |
3.nga. |
4.iol. |
5.nguenu. |
6.ngom-pum= 5-1.
7.ngom-miu= 5-2.
8.ngommag= 5-3.
9.nguenu-iol= 5-4.
10.to. |
Ashantee.248
1.tah. |
2.noo. |
3.sah. |
4.nah. |
5.taw. |
6.torata= 5 + 1.
7.toorifeenoo= 5 + 2.
8.toorifeessa= 5 + 3.
9.toorifeena= 5 + 4.
10.nopnoo. |
Basa.249
1.do. |
2.so. |
3.ta. |
4.hinye. |
5.hum. |
6.hum-le-do= 5 + 1.
7.hum-le-so= 5 + 2.
8.hum-le-ta= 5 + 3.
9.hum-le-hinyo= 5 + 4.
10.bla-bue. |
Jallonkas.250
1.kidding. |
2.fidding. |
3.sarra. |
4.nani. |
5.soolo. |
6.seni. |
7.soolo ma fidding= 5 + 2.
8.soolo ma sarra= 5 + 3.
9.soolo ma nani= 5 + 4.
10.nuff. |
Kru.
1.da-do. |
2.de-son. |
3.de-tan. |
4.de-nie. |
5.de-mu. |
6.dme-du= 5-1.
7.ne-son= [5] + 2.
8.ne-tan= [5] + 3.
9.sepadu= 10 − 1?
10.pua. |
Jaloffs.251
1.wean. |
2.yar. |
3.yat. |
4.yanet. |
5.judom. |
6.judom-wean= 5-1.
7.judom-yar= 5-2.
8.judom-yat= 5-3.
9.judom yanet= 5-4.
10.fook. |
Golo.252
1.mbali. |
2.bisi. |
3.bitta. |
4.banda. |
5.zonno. |
6.tsimmi tongbali= 5 + 1.
7.tsimmi tobisi= 5 + 2.
8.tsimmi tobitta= 5 + 3.
9.tsimmi to banda= 5 + 4.
10.nifo. |
Foulah.253
1.go. |
2.deeddee. |
3.tettee. |
4.nee. |
5.jouee. |
6.jego= 5-1.
7.jedeeddee= 5-2.
8.je-tettee= 5-3.
9.je-nee= 5-4.
10.sappo. |
Soussou.254
1.keren. |
2.firing. |
3.sarkan. |
4.nani. |
5.souli. |
6.seni. |
7.solo-fere= 5-2.
8.solo-mazarkan= 5 + 3.
9.solo-manani= 5 + 4.
10.fu. |
Bullom.255
1.bul. |
2.tin. |
3.ra. |
4.hyul. |
5.men. |
6.men-bul= 5-1.
7.men-tin= 5-2.
8.men-ra= 5-3.
9.men-hyul= 5-4.
10.won. |
Vei.256
1.dondo. |
2.fera. |
3.sagba. |
4.nani. |
5.soru. |
6.sun-dondo= 5-1.
7.sum-fera= 5-2.
8.sun-sagba= 5-3.
9.sun-nani= 5-4.
10.tan. |
Dinka.257
1.tok. |
2.rou. |
3.dyak. |
4.nuan. |
5.wdyets. |
6.wdetem= 5-1.
7.wderou= 5-2.
8.bet, bed= 5-3.
9.wdenuan= 5-4.
10.wtyer= 5 × 2.
Temne.
1.in. |
2.ran. |
3.sas. |
4.anle. |
5.tr-amat. |
6.tr-amat rok-in= 5 + 1.
7.tr-amat de ran= 5 + 2.
8.tr-amat re sas= 5 + 3.
9.tr-amat ro n-anle= 5 + 4.
10.tr-ofatr. |
Abaker.258
1.kili. |
2.bore. |
3.dotla. |
4.ashe. |
5.ini. |
6.im kili= 5-1.
7.im-bone= 5-2.
8.ini-dotta= 5-3.
9.tin ashe= 5-4.
10.chica. |
Bagrimma.259
1.kede. |
2.sab. |
3.muta. |
4.so. |
5.mi. |
6.mi-ga= 5 + 1.
7.tsidi. |
8.marta= 5 + 23.
9.do-so= [5] + 34
10.duk-keme. |
Papaa.260
1.depoo. |
2.auwi. |
3.ottong. |
4.enne. |
5.attong. |
6.attugo. |
7.atjuwe= [5] + 2.
8.attiatong= [5] + 3.
9.atjeenne= [5] + 4.
10.awo. |
Efik.261
1.kiet. |
2.iba. |
3.ita. |
4.inan. |
5.itiun. |
6.itio-kiet= 5-1.
7.itia-ba= 5-2.
8.itia-eta= 5-3.
9.osu-kiet= 10 − 1?
10.duup. |
Nupe.262
1.nini. |
2.gu-ba. |
3.gu-ta. |
4.gu-ni. |
5.gu-tsun. |
6.gu-sua-yin= 5 + 1.
7.gu-tua-ba= 5 + 2.
8.gu-tu-ta= 5 + 3.
9.gu-tua-ni= 5 + 4.
10.gu-wo. |
Mokko.263
1.kiä. |
2.iba. |
3.itta. |
4.inan. |
5.üttin. |
6.itjüekee= 5 + 1.
7.ittiaba= 5 + 2.
8.itteiata= 5 + 3.
9.huschukiet. |
10.büb. |
Kanuri.264
1.tilo. |
2.ndi. |
3.yasge. |
4.dege. |
5.ugu. |
6.arasge= 5 + 1.
7.tulur. |
8.wusge= 5 + 3.
9.legar. |
10.megu= 2 × 5.
Binin.265
1.bo. |
2.be. |
3.la. |
4.nin. |
5.tang. |
6.tahu= 5 + 1?
7.tabi= 5 + 2.
8.tara= 5 + 3.
9.ianin (tanin?)= 5 + 4?
10.te. |
Kredy.266
1.baia. |
2.rommu. |
3.totto. |
4.sosso. |
5.saya. |
6.yembobaia= [5] + 1.
7.yemborommu= [5] + 2.
8.yembototto= [5] + 3.
9.yembososso= [5] + 4.
10.puh. |
Herero.267
1.mue. |
2.vari. |
3.tatu. |
4.ne. |
5.tano. |
6.hambou-mue= [5] + 1.
7.hambou-vari= [5] + 2.
8.hambou-tatu= [5] + 3.
9.hambou-ne= [5] + 4.
10. |
Ki-Yau.268
1.jumo. |
2.wawiri. |
3.watatu. |
4.mcheche. |
5.msano. |
6.musano na jumo= 5 + 1.
7.musano na wiri= 5 + 2.
8.musano na watatu= 5 + 3.
9.musano na mcheche= 5 + 4.
10.ikumi. |
Fernando Po.269
1.muli. |
2.mempa. |
3.meta. |
4.miene. |
5.mimito. |
6.mimito na muli= 5 + 1.
7.mimito na mempa= 5 + 2.
8.mimito na meta= 5 + 3.
9.mimito na miene= 5 + 4.
10.miemieu= 5-5?
Ki-Nyassa
1.kimodzi. |
2.vi-wiri. |
3.vi-tatu. |
4.vinye. |
5.visano. |
6.visano na kimodzi= 5 + 1.
7.visano na vi-wiri= 5 + 2.
8.visano na vitatu= 5 + 3.
9.visano na vinye= 5 + 4.
10.chikumi. |
Balengue.270
1.guevoho. |
2.ibare. |
3.raro. |
4.inaï. |
5.itano. |
6.itano na guevoho= 5 + 1.
7.itano na ibare= 5 + 2.
8.itano na raro= 5 + 3.
9.itano na inaï= 5 + 4.
10.ndioum, or nai-hinaï. |
Kunama.271
1.ella. |
2.bare. |
3.sadde. |
4.salle. |
5.kussume. |
6.kon-t'-ella= hand 1.
7.kon-te-bare= hand 2.
8.kon-te-sadde= hand 3.
9.kon-te-salle= hand 4.
10.kol-lakada. |
Gola.272
1.ngoumou. |
2.ntie. |
3.ntaï. |
4.tina. |
5.nonon. |
6.diegoum= [5] + 1.
7.dientie= [5] + 2.
8.dietai= [5] + 3.
9.dectina= [5] + 4.
10.esia. |
Barea.273
1.doko |
2.arega. |
3.sane. |
4.sone. |
5.oita. |
6.data. |
7.dz-ariga= 5 + 2.
8.dis-sena= 5 + 3.
9.lefete-mada= without 10.
10.lefek. |
Matibani.274
1.mosa. |
2.pili. |
3.taru. |
4.teje. |
5.taru. |
6.tana mosa= 5-1.
7.tana pili= 5-2.
8.tana taru= 5-3.
9.loco. |
10.loco nakege. |
Bonzé.275
1.tan. |
2.vele. |
3.daba. |
4.nani. |
5.lolou. |
6.maïda= [5] + 1.
7.maïfile= [5] + 2.
8.maïshaba= [5] + 3.
9.maïnan= [5] + 4.
10.bou. |
Mpovi
1.moueta. |
2.bevali. |
3.betata. |
4.benaï. |
5.betani. |
6.betani moueta= 5-1.
7.betani bevali= 5-2.
8.betani betata= 5-3.
9.betani benai= 5-4.
10.nchinia. |
Triton's Bay, New Quinea.276
1.samosi. |
2.roueti. |
3.tourou. |
4.faat. |
5.rimi. |
6.rim-samosi= 5-1.
7.rim-roueti= 5-2.
8.rim-tourou= 5-3.
9.rim-faat= 5-4.
10.outsia. |
Ende, or Flores.277
1.sa. |
2.zua. |
3.telu. |
4.wutu. |
5.lima= hand.
6.lima-sa= 5-1, or hand 1.
7.lima-zua= 5-2.
8.rua-butu= 2 × 4?
9.trasa= [10] − 1?
10.sabulu. |
Mallicolo.278
1.tseekaee. |
2.ery. |
3.erei. |
4.ebats. |
5.ereem. |
6.tsookaee= [5] + 1.
7.gooy= [5] + 2.
8.hoorey= [5] + 3.
9.goodbats= [5] + 4.
10.senearn. |
Ebon, Marshall Islands.279
1.iuwun. |
2.drud. |
3.chilu. |
4.emer. |
5.lailem. |
6.chilchinu= 5 + 1.
7.chilchime= 5 + 2.
8.twalithuk= [10] − 2.
9.twahmejuwou= [10] − 1.
10.iungou. |
Uea.280—[another dialect.]
1.hacha. |
2.lo. |
3.kuun. |
4.thack. |
5.thabumb. |
6.lo-acha= 2d 1.
7.lo-alo= 2d 2.
8.lo-kuun= 2d 3.
9.lo-thack= 2d 4.
10.lebenetee. |
Isle of Pines.281
1.ta. |
2.bo. |
3.beti. |
4.beu. |
5.ta-hue. |
6.no-ta= 2d 1.
7.no-bo= 2d 2.
8.no-beti= 2d 3.
9.no-beu= 2d 4.
10.de-kau. |
Ureparapara, Banks Islands.282
1.vo towa. |
2.vo ro. |
3.vo tol. |
4.vo vet. |
5.teveliem= 1 hand.
6.leve jea= other 1.
7.leve ro= other 2.
8.leve tol= other 3.
9.leve vet= other 4.
10.sanowul= 2 sets.
Mota, Banks Islands.282
1.tuwale. |
2.nirua. |
3.nitol. |
4.nivat. |
5.tavelima= 1 hand.
6.laveatea= other 1.
7.lavearua= other 2.
8.laveatol= other 3.
9.laveavat= other 4.
10.sanavul= 2 sets.
New Caledonia.283
1.parai. |
2.paroo. |
3.parghen. |
4.parbai. |
5.panim. |
6.panim-gha= 5-1.
7.panim-roo= 5-2.
8.panim-ghen= 5-3.
9.panim-bai= 5-4.
10.parooneek. |
Yengen, New Cal.284
1.hets. |
2.heluk. |
3.heyen. |
4.pobits. |
5.nim= hand.
6.nim-wet= 5-1.
7.nim-weluk= 5-2.
8.nim-weyen= 5-3.
9.nim-pobit= 5-4.
10.pain-duk. |
Aneiteum.285
1.ethi. |
2.ero. |
3.eseik. |
4.manohwan. |
5.nikman. |
6.nikman cled et ethi= 5 + 1.
7.nikman cled et oro= 5 + 2.
8.nikman cled et eseik= 5 + 3.
9.nikman cled et manohwan= 5 + 4.
10.nikman lep ikman= 5 + 5.
Tanna
1.riti. |
2.karu. |
3.kahar. |
4.kefa. |
5.krirum. |
6.krirum riti= 5-1.
7.krirum karu= 5-2.
8.krirum kahar?= 5-3.
9.krirum kefa?= 5-4.
10.—— |
Eromanga
1.sai. |
2.duru. |
3.disil. |
4.divat. |
5.siklim= 1 hand.
6.misikai= other 1?
7.siklim naru= 5-2.
8.siklim disil= 5-3.
9.siklim mindivat= 5 + 4.
10.narolim= 2 hands.
Fate, New Heb.286
1.iskei. |
2.rua. |
3.tolu. |
4.bate. |
5.lima= hand.
6.la tesa= other 1.
7.la rua= other 2.
8.la tolu= other 3.
9.la fiti= other 4.
10.relima= 2 hands.
Api, New Heb.
1.tai. |
2.lua. |
3.tolu. |
4.vari. |
5.lima= hand.
6.o rai= other 1.
7.o lua= other 2.
8.o tolo= other 3.
9.o vari= other 4.
10.lua lima= 2 hands.
Sesake, New Heb.
1.sikai. |
2.dua. |
3.dolu. |
4.pati. |
5.lima= hand.
6.la tesa= other 1.
7.la dua= other 2.
8.la dolu= other 3.
9.lo veti= other 4.
10.dua lima= 2 hands.
Pama, New Heb.
1.tai. |
2.e lua. |
3.e tolu. |
4.e hati. |
5.e lime= hand.
6.a hitai= other 1.
7.o lu= other 2.
8.o tolu= other 3.
9.o hati= other 4.
10.ha lua lim= 2 hands
Aurora, New Heb.
1.tewa. |
2.i rua. |
3.i tol. |
4.i vat. |
5.tavalima= 1 hand.
6.lava tea= other 1.
7.lava rua= other 2.
8.lava tol= other 3.
9.la vat= other 4.
10.sanwulu= two sets.
Tobi.287
1.yat. |
2.glu. |
3.ya. |
4.uan. |
5.yanim= 1 hand.
6.yawor= other 1.
7.yavic= other 2.
8.yawa= other 3.
9.yatu= other 4.
10.yasec. |
Palm Island.288
1.yonkol. |
2.yakka. |
3.tetjora. |
4.tarko. |
5.yonkol mala= 1 hand.
Jajowerong, Victoria.288
1.kiarp. |
2.bulaits. |
3.bulaits kiarp= 2-1.
4.bulaits bulaits= 2-2.
5.kiarp munnar= 1 hand.
6.bulaits bulaits bulaits= 2-2-2.
10.bulaits munnar= 2 hands.
The last two scales deserve special notice. They are
Australian scales, and the former is strongly binary, as
are so many others of that continent. But both show
an incipient quinary tendency in their names for 5 and
10.
Cambodia.289
1.muy. |
2.pir. |
3.bey. |
4.buon. |
5.pram. |
6.pram muy= 5-1.
7.pram pil= 5-2.
8.pram bey= 5-3.
9.pram buon= 5-4.
10.dap. |
Tschukschi.290
1.inen. |
2.nirach. |
3.n'roch. |
4.n'rach. |
5.miligen= hand.
6.inen miligen= 1-5.
7.nirach miligen= 2-5.
8.anwrotkin. |
9.chona tsinki. |
10.migitken= both hands.
Kottisch291
1.hutsa. |
2.ina. |
3.tona. |
4.sega. |
5.chega. |
6.chelutsa= 5 + 1.
7.chelina= 5 + 2.
8.chaltona= 5 + 3.
9.tsumnaga= 10 − 1.
10.haga. |
Eskimo of N.-W. Alaska.292
1.a towshek. |
2.hipah, or malho. |
3.pingishute. |
4.sesaimat. |
5.talema. |
6.okvinile, or ahchegaret= another 1?
7.talema-malronik= 5-two of them.
8.pingishu-okvingile= 2d 3?
9.kolingotalia= 10 − 1?
10.koleet. |
Kamtschatka, South.293
1.dischak. |
2.kascha. |
3.tschook. |
4.tschaaka. |
5.kumnaka. |
6.ky'lkoka. |
7.itatyk= 2 + 5.
8.tschookotuk= 3 + 5.
9.tschuaktuk= 4 + 5.
10.kumechtuk= 5 + 5.
Aleuts294
1.ataqan. |
2.aljak. |
3.qankun. |
4.sitsin. |
5.tsan= my hand.
6.atun= 1 + 5.
7.ulun= 2 + 5.
8.qamtsin= 3 + 5.
9.sitsin= 4 + 5.
10.hatsiq. |
Tchiglit, Mackenzie R.295
1.ataotçirkr. |
2.aypak, or malloerok. |
3.illaak, or piñatcut. |
4.tçitamat. |
5.tallemat. |
6.arveneloerit. |
7.arveneloerit-aypak= 5 + 2.
8.arveneloerit-illaak= 5 + 3.
9.arveneloerit-tçitamat= 5 + 4.
10.krolit. |
Sahaptin (Nez Perces).296
1.naks. |
2.lapit. |
3.mitat. |
4.pi-lapt= 2 × 2.
5.pachat. |
6.oi-laks= [5] + 1.
7.oi-napt= [5] + 2.
8.oi-matat= [5] + 3.
9.koits. |
10.putimpt. |
Greenland.297
1.atauseq. |
2.machdluq. |
3.pinasut. |
4.sisamat |
5.tadlimat. |
6.achfineq-atauseq= other hand 1.
7.achfineq-machdluq= other hand 2.
8.achfineq-pinasut= other hand 3.
9.achfineq-sisamat= other hand 4.
10.qulit. |
11.achqaneq-atauseq= first foot 1.
12.achqaneq-machdluq= first foot 2.
13.achqaneq-pinasut= first foot 3.
14.achqaneq-sisamat= first foot 4.
15.achfechsaneq? |
16.achfechsaneq-atauseq= other foot 1.
17.achfechsaneq-machdlup= other foot 2.
18.achfechsaneq-pinasut= other foot 3.
19.achfechsaneq-sisamat= other foot 4.
20.inuk navdlucho= a man ended.
Up to this point the Greenlander's scale is almost
purely quinary. Like those of which mention was made
at the beginning of this chapter, it persists in progressing
by fives until it reaches 20, when it announces a
new base, which shows that the system will from now
on be vigesimal. This scale is one of the most interesting
of which we have any record, and will be
noticed again in the next chapter. In many respects
it is like the scale of the Point Barrow Eskimo, which
was given early in Chapter III. The Eskimo languages
are characteristically quinary-vigesimal in their number
systems, but few of them present such perfect
examples of that method of counting as do the two
just mentioned.
Chippeway.298
1.bejig. |
2.nij. |
3.nisswi. |
4.niwin. |
5.nanun. |
6.ningotwasswi= 1 again?
7.nijwasswi= 2 again?
8.nishwasswi= 3 again?
9.jangasswi= 4 again?
10.midasswi= 5 again.
Massachusetts.299
1.nequt. |
2.neese. |
3.nish. |
4.yaw. |
5.napanna= on one side, i.e. 1 hand.
6.nequttatash= 1 added.
7.nesausuk= 2 again?
8.shawosuk= 3 again?
9.pashoogun= it comes near, i.e. to 10.
10.puik. |
Ojibwa of Chegoimegon.300
1.bashik. |
2.neensh. |
3.niswe. |
4.newin. |
5.nanun. |
6.ningodwaswe= 1 again?
7.nishwaswe= 2 again?
8.shouswe= 3 again?
9.shangaswe= 4 again?
10.medaswe= 5 again?
Ottawa.
1.ningotchau. |
2.ninjwa. |
3.niswa. |
4.niwin. |
5.nanau. |
6.ningotwaswi= 1 again?
7.ninjwaswi= 2 again?
8.nichwaswi= 3 again?
9.shang. |
10.kwetch. |
Delaware.
1.n'gutti. |
2.niskha. |
3.nakha. |
4.newa. |
5.nalan [akin to palenach, hand]. |
6.guttash= 1 on the other side.
7.nishash= 2 on the other side.
8.khaash= 3 on the other side.
9.peshgonk= coming near.
10.tellen= no more.
Shawnoe.
1.negote. |
2.neshwa. |
3.nithuie. |
4.newe. |
5.nialinwe= gone.
6.negotewathwe= 1 further.
7.neshwathwe= 2 further.
8.sashekswa= 3 further?
9.chakatswe [akin to chagisse, “used up”]. |
10.metathwe= no further.
Micmac.301
1.naiookt.
2.tahboo.
3.seest.
4.naioo.
5.nahn.
6.usoo-cum.
7.eloo-igunuk.
8.oo-gumoolchin.
9.pescoonaduk.
10.mtlin.
One peculiarity of the Micmac numerals is most noteworthy.
The numerals are real verbs, instead of adjectives,
or, as is sometimes the case, nouns. They are
conjugated through all the variations of mood, tense, person,
and number. The forms given above are not those
that would be used in counting, but are for specific use,
being varied according to the thought it was intended
to express. For example,
naiooktaich = there is 1, is
present tense;
naiooktaichcus, there was 1, is imperfect;
and
encoodaichdedou, there will be 1, is future.
The variation in person is shown by the following
inflection:
Present Tense.
1st pers.tahboosee-ek= there are 2 of us.
2d pers.tahboosee-yok= there are 2 of you.
3d pers.tahboo-sijik= there are 2 of them.
Imperfect Tense.
1st pers.tahboosee-egup= there were 2 of us.
2d pers.tahboosee-yogup= there were 2 of you.
3d pers.tahboosee-sibunik= there were 2 of them.
Future Tense.
3d pers.tahboosee-dak= there will be 2 of them, etc.
The negative form is also comprehended in the list
of possible variations. Thus,
tahboo-seekw, there are not
2 of them;
mah tahboo-seekw, there will not be 2 of
them; and so on, through all the changes which the conjugation
of the verb permits.
Old Algonquin.
1.peygik. |
2.ninsh. |
3.nisswey. |
4.neyoo. |
5.nahran= gone.
6.ningootwassoo= 1 on the other side.
7.ninshwassoo= 2 on the other side.
8.nisswasso= 3 on the other side.
9.shangassoo [akin to chagisse, “used up”]. |
10.mitassoo= no further.
Omaha.
1.meeachchee. |
2.nomba. |
3.rabeenee. |
4.tooba. |
5.satta= hand, i.e. all the fingers turned down.
6.shappai= 1 more.
7.painumba= fingers 2.
8.pairabeenee= fingers 3.
9.shonka= only 1 finger (remains).
10.kraibaira= unbent.302
Choctaw.
1.achofee. |
2.tuklo. |
3.tuchina. |
4.ushta. |
5.tahlape= the first hand ends.
6.hanali. |
7.untuklo= again 2.
8.untuchina= again 3.
9.chokali= soon the end; i.e. next the last.
10.pokoli. |
Caddoe.
1.kouanigh. |
2.behit. |
3.daho. |
4.hehweh. |
5.dihsehkon. |
6.dunkeh. |
7.bisekah= 5 + 2.
8.dousehka= 5 + 3.
9.hehwehsehka= 4 + hand.
10.behnehaugh. |
Chippeway.
1.payshik. |
2.neesh. |
3.neeswoy. |
4.neon. |
5.naman= gone.
6.nequtwosswoy= 1 on the other side.
7.neeshswosswoy= 2 on the other side.
8.swoswoy= 3 on the other side?
9.shangosswoy [akin to chagissi, “used up”]. |
10.metosswoy= no further.
Adaize.
1.nancas. |
2.nass. |
3.colle. |
4.tacache. |
5.seppacan. |
6.pacanancus= 5 + 1.
7.pacaness= 5 + 2.
8.pacalcon= 5 + 3.
9.sickinish= hands minus?
10.neusne. |
Pawnee.
1.askoo. |
2.peetkoo. |
3.touweet. |
4.shkeetiksh. |
5.sheeooksh= hands half.
6.sheekshabish= 5 + 1.
7.peetkoosheeshabish= 2 + 5.
8.touweetshabish= 3 + 5.
9.looksheereewa= 10 − 1.
10.looksheeree= 2d 5?
Minsi.
1.gutti. |
2.niskha. |
3.nakba. |
4.newa. |
5.nulan= gone?
6.guttash= 1 added.
7.nishoash= 2 added.
8.khaash= 3 added.
9.noweli. |
10.wimbat. |
Konlischen.
1.tlek. |
2.tech. |
3.nezk. |
4.taakun. |
5.kejetschin. |
6.klet uschu= 5 + 1.
7.tachate uschu= 5 + 2.
8.nesket uschu= 5 + 3.
9.kuschok= 10 − 1?
10.tschinkat. |
Tlingit.303
1.tlek. |
2.deq. |
3.natsk. |
4.dak'on= 2d 2.
5.kedjin= hand.
6.tle durcu= other 1.
7.daqa durcu= other 2.
8.natska durcu= other 3.
9.gocuk. |
10.djinkat= both hands.
Rapid, or Fall, Indians.
1.karci. |
2.neece. |
3.narce. |
4.nean. |
5.yautune. |
6.neteartuce= 1 over?
7.nesartuce= 2 over?
8.narswartuce= 3 over?
9.anharbetwartuce= 4 over?
10.mettartuce= no further?
Heiltsuk.304
1.men. |
2.matl. |
3.yutq. |
4.mu. |
5.sky'a. |
6.katla. |
7.matlaaus= other 2?
8.yutquaus= other 3?
9.mamene= 10 − 1.
10.aiky'as. |
Nootka.305
1.nup. |
2.atla. |
3.katstsa. |
4.mo. |
5.sutca. |
6.nopo= other 1?
7.atlpo= other 2?
8.atlakutl= 10 − 2.
9.ts'owakutl= 10 − 1.
10.haiu. |
Tsimshian.306
1.gyak. |
2.tepqat. |
3.guant. |
4.tqalpq. |
5.kctonc (from anon, hand). |
6.kalt= 2d 1.
7.t'epqalt= 2d 2.
8.guandalt= 2d 3?
9.kctemac. |
10.gy'ap. |
Bilqula.306
1.(s)maotl. |
2.tlnos. |
3.asmost. |
4.mos. |
5.tsech. |
6.tqotl= 2d 1?
7.nustlnos= 2d 2?
8.k'etlnos= 2 × 4.
9.k'esman. |
10.tskchlakcht. |
Molele.307
1.mangu. |
2.lapku. |
3.mutka. |
4.pipa. |
5.pika. |
6.napitka= 1 + 5.
7.lapitka= 2 + 5.
8.mutpitka= 3 + 5.
9.laginstshiatkus. |
10.nawitspu. |
Waiilatpu.308
1.na. |
2.leplin. |
3.matnin. |
4.piping. |
5.tawit. |
6.noina= [5] + 1.
7.noilip= [5] + 2.
8.noimat= [5] + 3.
9.tanauiaishimshim. |
10.ningitelp. |
Lutuami.307
1.natshik. |
2.lapit. |
3.ntani. |
4.wonip. |
5.tonapni. |
6.nakskishuptane= 1 + 5.
7.tapkishuptane= 2 + 5.
8.ndanekishuptane= 3 + 5.
9.natskaiakish= 10 − 1.
10.taunip. |
Saste (Shasta).309
1.tshiamu. |
2.hoka. |
3.hatski. |
4.irahaia. |
5.etsha. |
6.tahaia. |
7.hokaikinis= 2 + 5.
8.hatsikikiri= 3 + 5.
9.kirihariki-ikiriu. |
10.etsehewi. |
Cahuillo.310
1.supli. |
2.mewi. |
3.mepai. |
4.mewittsu. |
5.nomekadnun. |
6.kadnun-supli= 5-1.
7.kan-munwi= 5-2.
8.kan-munpa= 5-3.
9.kan-munwitsu= 5-4.
10.nomatsumi. |
Timukua.311
1.yaha. |
2.yutsa. |
3.hapu. |
4.tseketa. |
5.marua. |
6.mareka= 5 + 1
7.pikitsa= 5 + 2
8.pikinahu= 5 + 3
9.peke-tsaketa= 5 + 4
10.tuma. |
Otomi312
1.nara. |
2.yocho. |
3.chiu. |
4.gocho. |
5.kuto. |
6.rato= 1 + 5.
7.yoto= 2 + 5.
8.chiato= 3 + 5.
9.guto= 4 + 5.
10.reta. |
Tarasco.313
1.ma. |
2.dziman. |
3.tanimo. |
4.tamu. |
5.yumu. |
6.kuimu. |
7.yun-dziman= [5] + 2.
8.yun-tanimo= [5] + 3.
9.yun-tamu= [5] + 4.
10.temben. |
Matlaltzincan.314
1.indawi. |
2.inawi. |
3.inyuhu. |
4.inkunowi. |
5.inkutaa. |
6.inda-towi= 1 + 5.
7.ine-towi= 2 + 5.
8.ine-ukunowi= 2-4.
9.imuratadahata= 10 − 1?
10.inda-hata. |
Cora.315
1.ceaut. |
2.huapoa. |
3.huaeica. |
4.moacua. |
5.anxuvi. |
6.a-cevi= [5] + 1.
7.a-huapoa= [5] + 2.
8.a-huaeica= [5] + 3.
9.a-moacua= [5] + 4.
10.tamoamata (akin to moamati, “hand”). |
Aymara.316
1.maya. |
2.paya. |
3.kimsa. |
4.pusi. |
5.piska. |
6.tsokta. |
7.pa-kalko= 2 + 5.
8.kimsa-kalko= 3 + 5.
9.pusi-kalko= 4 + 5.
10.tunka. |
Caribs of Essequibo, Guiana.317
1.oween. |
2.oko. |
3.oroowa. |
4.oko-baimema. |
5.wineetanee= 1 hand.
6.owee-puimapo= 1 again?
7.oko-puimapo= 2 again?
8.oroowa-puimapo= 3 again?
9.oko-baimema-puimapo= 4 again?
10.oween-abatoro. |
Carib.318 (Roucouyenne?)
1.aban, amoin. |
2.biama. |
3.eleoua. |
4.biam-bouri= 2 again?
5.ouacabo-apourcou-aban-tibateli. |
6.aban laoyagone-ouacabo-apourcou. |
7.biama laoyagone-ouacabo-apourcou. |
8.eleoua laoyagone-ouacabo-apourcou. |
9.—— |
10.chon noucabo. |
It is unfortunate that the meanings of these remarkable
numerals cannot be given. The counting is evidently
quinary, but the terms used must have been purely
descriptive expressions, having their origin undoubtedly
in certain gestures or finger motions. The numerals
obtained from this region, and from the tribes to the
south and east of the Carib country, are especially rich
in digital terms, and an analysis of the above numerals
would probably show clearly the mental steps through
which this people passed in constructing the rude scale
which served for the expression of their ideas of number.
Kiriri.319
1.biche. |
2.watsani. |
3.watsani dikie. |
4.sumara oroba. |
5.mi biche misa= 1 hand.
6.mirepri bu-biche misa sai. |
7.mirepri watsani misa sai. |
8.mirepri watsandikie misa sai. |
9.mirepri sumara oraba sai. |
10.mikriba misa sai= both hands.
Cayubaba320
1.pebi. |
2.mbeta. |
3.kimisa. |
4.pusi. |
5.pisika. |
6.sukuta. |
7.pa-kaluku= 2 again?
8.kimisa-kaluku= 3 again?
9.pusu-kaluku= 4 again?
10.tunka. |
Sapibocona320
1.karata. |
2.mitia. |
3.kurapa. |
4.tsada. |
5.maidara (from arue, hand). |
6.karata-rirobo= 1 hand with.
7.mitia-rirobo= 2 hand with.
8.kurapa-rirobo= 3 hand with.
9.tsada-rirobo= 4 hand with.
10.bururutse= hand hand.
Ticuna.321
1.hueih. |
2.tarepueh. |
3.tomepueh. |
4.aguemoujih |
5.hueamepueh. |
6.naïmehueapueh= 5 + 1.
7.naïmehueatareh= 5 + 2.
8.naïmehueatameapueh= 5 + 3.
9.gomeapueh= 10 − 1.
10.gomeh. |
Yanua.322
1.tckini. |
2.nanojui. |
3.munua. |
4.naïrojuino= 2d 2.
5.tenaja. |
6.teki-natea= 1 again?
7.nanojui-natea= 2 again?
8.munua-natea= 3 again?
9.naïrojuino-natea= 4 again?
10.huijejuino= 2 × 5?
The foregoing examples will show with considerable
fulness the wide dispersion of the quinary scale. Every
part of the world contributes its share except Europe,
where the only exceptions to the universal use of the
decimal system are the half-dozen languages, which still
linger on its confines, whose number base is the vigesimal.
Not only is there no living European tongue
possessing a quinary number system, but no trace of
this method of counting is found in any of the numerals
of the earlier forms of speech, which have now
become obsolete. The only possible exceptions of which
I can think are the Greek
πεμπάζειν, to count by fives,
and a few kindred words which certainly do hint at a
remote antiquity in which the ancestors of the Greeks
counted on their fingers, and so grouped their units
into fives. The Roman notation, the familiar I., II., III.,
IV. (originally IIII.), V., VI., etc., with equal certainty
suggests quinary counting, but the Latin language
contains no vestige of anything of the kind, and the
whole range of Latin literature is silent on this point,
though it contains numerous references to finger counting.
It is quite within the bounds of possibility that
the prehistoric nations of Europe possessed and used a
quinary numeration. But of these races the modern
world knows nothing save the few scanty facts that
can be gathered from the stone implements which have
now and then been brought to light. Their languages
have perished as utterly as have the races themselves,
and speculation concerning them is useless. Whatever
their form of numeration may have been, it has left
no perceptible trace on the languages by which they
were succeeded. Even the languages of northern and
central Europe which were contemporary with the
Greek and Latin of classical times have, with the
exception of the Celtic tongues of the extreme North-west,
left behind them but meagre traces for the
modern student to work on. We presume that the
ancient Gauls and Goths, Huns and Scythians, and
other barbarian tribes had the same method of numeration
that their descendants now have; and it is a
matter of certainty that the decimal scale was, at that
time, not used with the universality which now obtains;
but wherever the decimal was not used, the universal
method was vigesimal; and that the quinary ever had
anything of a foothold in Europe is only to be guessed
from its presence to-day in almost all of the other
corners of the world.
From the fact that the quinary is that one of the
three natural scales with the smallest base, it has been
conjectured that all tribes possess, at some time in
their history, a quinary numeration, which at a later
period merges into either the decimal or the vigesimal,
and thus disappears or forms with one of the latter a
mixed system.323 In support of this theory it is urged
that extensive regions which now show nothing but
decimal counting were, beyond all reasonable doubt,
quinary. It is well known, for example, that the decimal
system of the Malays has spread over almost the
entire Polynesian region, displacing whatever native
scales it encountered. The same phenomenon has been
observed in Africa, where the Arab traders have disseminated
their own numeral system very widely, the
native tribes adopting it or modifying their own scales
in such a manner that the Arab influence is detected
without difficulty.
In view of these facts, and of the extreme readiness
with which a tribe would through its finger counting
fall into the use of the quinary method, it does not at
first seem improbable that the quinary was the original
system. But an extended study of the methods of
counting in vogue among the uncivilized races of all
parts of the world has shown that this theory is entirely
untenable. The decimal scale is no less simple
in its structure than the quinary; and the savage, as
he extends the limit of his scale from 5 to 6, may call
his new number 5-1, or, with equal probability, give it
an entirely new name, independent in all respects of
any that have preceded it. With the use of this new
name there may be associated the conception of “5
and 1 more”; but in such multitudes of instances the
words employed show no trace of any such meaning, that
it is impossible for any one to draw, with any degree
of safety, the inference that the signification was originally
there, but that the changes of time had wrought
changes in verbal form so great as to bury it past the
power of recovery. A full discussion of this question
need not be entered upon here. But it will be of interest
to notice two or three numeral scales in which
the quinary influence is so faint as to be hardly discernible.
They are found in considerable numbers
among the North American Indian languages, as may
be seen by consulting the vocabularies that have been
prepared and published during the last half century.324
From these I have selected the following, which are
sufficient to illustrate the point in question:
Quappa.
1.milchtih.
2.nonnepah.
3.dahghenih.
4.tuah.
5.sattou.
6.schappeh.
7.pennapah.
8.pehdaghenih.
9.schunkkah.
10.gedeh bonah.
Terraba.325
1.krara.
2.krowü.
3.krom miah.
4.krob king.
5.krasch kingde.
6.terdeh.
7.kogodeh.
8.kwongdeh.
9.schkawdeh.
10.dwowdeh.
Mohican
1.ngwitloh.
2.neesoh.
3.noghhoh.
4.nauwoh.
5.nunon.
6.ngwittus.
7.tupouwus.
8.ghusooh.
9.nauneeweh.
10.mtannit.
In the Quappa scale 7 and 8 appear to be derived
from 2 and 3, while 6 and 9 show no visible trace
of kinship with 1 and 4. In Mohican, on the other
hand, 6 and 9 seem to be derived from 1 and 4, while
7 and 8 have little or no claim to relationship with
2 and 3. In some scales a single word only is found
in the second quinate to indicate that 5 was originally
the base on which the system rested. It is hardly to
be doubted, even, that change might affect each and
every one of the numerals from 5 to 10 or 6 to 9, so
that a dependence which might once have been easily
detected is now unrecognizable.
But if this is so, the natural and inevitable question
follows—might not this have been the history of all
numeral scales now purely decimal? May not the
changes of time have altered the compounds which
were once a clear indication of quinary counting, until
no trace remains by which they can be followed back
to their true origin? Perhaps so. It is not in the
least degree probable, but its possibility may, of course,
be admitted. But even then the universality of quinary
counting for primitive peoples is by no means
established. In Chapter II, examples were given of races
which had no number base. Later on it was observed
that in Australia and South America many tribes used
2 as their number base; in some cases counting on past
5 without showing any tendency to use that as a new
unit. Again, through the habit of counting upon the
finger joints, instead of the fingers themselves, the use
of 3 as a base is brought into prominence, and 6 and
9 become 2 threes and 3 threes, respectively, instead of
5 + 1 and 5 + 4. The same may be noticed of 4. Counting
by means of his fingers, without including the
thumbs, the savage begins by dividing into fours instead
of fives. Traces of this form of counting are somewhat
numerous, especially among the North American aboriginal
tribes. Hence the quinary form of counting,
however widespread its use may be shown to be, can
in no way be claimed as the universal method of any
stage of development in the history of mankind.
In the vast majority of cases, the passage from the
base to the next succeeding number in any scale, is
clearly defined. But among races whose intelligence is
of a low order, or—if it be permissible to express
it in this way—among races whose number sense is
feeble, progression from one number to the next is not
always in accordance with any well-defined law. After
one or two distinct numerals the count may, as in the
case of the Veddas and the Andamans, proceed by finger
pantomime and by the repetition of the same word.
Occasionally the same word is used for two successive
numbers, some gesture undoubtedly serving to distinguish
the one from the other in the savage's mind.
Examples of this are not infrequent among the forest
tribes of South America. In the Tariana dialect 9
and 10 are expressed by the same word, paihipawalianuda;
in Cobeu, 8 and 9 by pepelicoloblicouilini; in
Barre, 4, 5, and 9 by ualibucubi.326 In other languages the
change from one numeral to the next is so slight that
one instinctively concludes that the savage is forming
in his own mind another, to him new, numeral immediately
from the last. In such cases the entire number
system is scanty, and the creeping hesitancy with which
progress is made is visible in the forms which the numerals
are made to take. A single illustration or two
of this must suffice; but the ones chosen are not isolated
cases. The scale of the Macunis,327 one of the numerous
tribes of Brazil, is
1.pocchaenang. |
2.haihg. |
3.haigunhgnill. |
4.haihgtschating. |
5.haihgtschihating= another 4?
6.hathig-stchihathing= 2-4?
7.hathink-tschihathing= 2-5?
8.hathink-tschihating= 2 × 4?
The complete absence of—one is tempted to say—any
rhyme or reason from this scale is more than
enough to refute any argument which might tend to
show that the quinary, or any other scale, was ever the
sole number scale of primitive man. Irregular as this is,
the system of the Montagnais fully matches it, as the
subjoined numerals show:328
1.inl'are. |
2.nak'e. |
3.t'are. |
4.dinri. |
5.se-sunlare. |
6.elkke-t'are= 2 × 3.
7.t'a-ye-oyertan= 10 − 3,
or inl'as dinri= 4 + 3? |
8.elkke-dinri= 2 × 4.
9.inl'a-ye-oyertan= 10 − 1.
10.onernan. |
The Vigesimal System.
In its ordinary development the quinary system is
almost sure to merge into either the decimal or the
vigesimal system, and to form, with one or the other
or both of these, a mixed system of counting. In
Africa, Oceanica, and parts of North America, the
union is almost always with the decimal scale; while
in other parts of the world the quinary and the vigesimal
systems have shown a decided affinity for each
other. It is not to be understood that any geographical
law of distribution has ever been observed which
governs this, but merely that certain families of races
have shown a preference for the one or the other
method of counting. These families, disseminating
their characteristics through their various branches,
have produced certain groups of races which exhibit
a well-marked tendency, here toward the decimal, and
there toward the vigesimal form of numeration. As
far as can be ascertained, the choice of the one or the
other scale is determined by no external circumstances,
but depends solely on the mental characteristics of
the tribes themselves. Environment does not exert any
appreciable influence either. Both decimal and vigesimal
numeration are found indifferently in warm and in
cold countries; in fruitful and in barren lands; in
maritime and in inland regions; and among highly
civilized or deeply degraded peoples.
Whether or not the principal number base of any
tribe is to be 20 seems to depend entirely upon a single
consideration; are the fingers alone used as an aid
to counting, or are both fingers and toes used? If
only the fingers are employed, the resulting scale must
become decimal if sufficiently extended. If use is made
of the toes in addition to the fingers, the outcome must
inevitably be a vigesimal system. Subordinate to either
one of these the quinary may and often does appear.
It is never the principal base in any extended system.
To the statement just made respecting the origin of
vigesimal counting, exception may, of course, be taken.
In the case of numeral scales like the Welsh, the Nahuatl,
and many others where the exact meanings of the
numerals cannot be ascertained, no proof exists that
the ancestors of these peoples ever used either finger or
toe counting; and the sweeping statement that any
vigesimal scale is the outgrowth of the use of these
natural counters is not susceptible of proof. But so
many examples are met with in which the origin is
clearly of this nature, that no hesitation is felt in putting
the above forward as a general explanation for the
existence of this kind of counting. Any other origin
is difficult to reconcile with observed facts, and still
more difficult to reconcile with any rational theory of
number system development. Dismissing from consideration
the quinary scale, let us briefly examine once
more the natural process of evolution through which
the decimal and the vigesimal scales come into being.
After the completion of one count of the fingers the
savage announces his result in some form which definitely
states to his mind the fact that the end of a well-marked
series has been reached. Beginning again, he
now repeats his count of 10, either on his own fingers
or on the fingers of another. With the completion of
the second 10 the result is announced, not in a new
unit, but by means of a duplication of the term already
used. It is scarcely credible that the unit unconsciously
adopted at the termination of the first count
should now be dropped, and a new one substituted in
its place. When the method here described is employed,
20 is not a natural unit to which higher numbers
may be referred. It is wholly artificial; and it
would be most surprising if it were adopted. But if
the count of the second 10 is made on the toes in
place of the fingers, the element of repetition which
entered into the previous method is now wanting. Instead
of referring each new number to the 10 already
completed, the savage is still feeling his way along,
designating his new terms by such phrases as “1 on
the foot,” “2 on the other foot,” etc. And now, when
20 is reached, a single series is finished instead of a
double series as before; and the result is expressed in
one of the many methods already noticed—“one man,”
“hands and feet,” “the feet finished,” “all the fingers
of hands and feet,” or some equivalent formula. Ten
is no longer the natural base. The number from which
the new start is made is 20, and the resulting scale is
inevitably vigesimal. If pebbles or sticks are used
instead of fingers, the system will probably be decimal.
But back of the stick and pebble counting the 10 natural
counters always exist, and to them we must always
look for the origin of this scale.
In any collection of the principal vigesimal number
systems of the world, one would naturally begin with
those possessed by the Celtic races of Europe. These
races, the earliest European peoples of whom we have
any exact knowledge, show a preference for counting
by twenties, which is almost as decided as that manifested
by Teutonic races for counting by tens. It has
been conjectured by some writers that the explanation
for this was to be found in the ancient commercial
intercourse which existed between the Britons and the
Carthaginians and Phœnicians, whose number systems
showed traces of a vigesimal tendency. Considering
the fact that the use of vigesimal counting was universal
among Celtic races, this explanation is quite
gratuitous. The reason why the Celts used this method
is entirely unknown, and need not concern investigators
in the least. But the fact that they did use it
is important, and commands attention. The five Celtic
languages, Breton, Irish, Welsh, Manx, and Gaelic, contain
the following well-defined vigesimal scales. Only
the principal or characteristic numerals are given, those
being sufficient to enable the reader to follow intelligently
the growth of the systems. Each contains the
decimal element also, and is, therefore, to be regarded
as a mixed decimal-vigesimal system.
Irish.329
10.deic. |
20.fice. |
30.triocad= 3-10
40.da ficid= 2-20.
50.caogad= 5-10.
60.tri ficid= 3-20.
70.reactmoga= 7-10.
80.ceitqe ficid= 4-20.
90.nocad= 9-10.
100.cead. |
1000.mile. |
Gaelic.330
10.deich. |
20.fichead. |
30.deich ar fichead= 10 + 20.
40.da fhichead= 2-20.
50.da fhichead is deich= 40 + 10.
60.tri fichead= 3-20.
70.tri fichead is deich= 60 + 10.
80.ceithir fichead= 4-20.
90.ceithir fichead is deich= 80 + 10.
100.ceud. |
1000.mile. |
Welsh.331
10.deg. |
20.ugain. |
30.deg ar hugain= 10 + 20.
40.deugain= 2-20.
50.deg a deugain= 10 + 40.
60.trigain= 3-20.
70.deg a thrigain= 10 + 60.
80.pedwar ugain= 4-20.
90.deg a pedwar ugain= 80 + 10.
100.cant. |
Manx.332
10.jeih. |
20.feed. |
30.yn jeih as feed= 10 + 20.
40.daeed= 2-20.
50.jeih as daeed= 10 + 40.
60.three-feed= 3-20.
70.three-feed as jeih= 60 + 10.
80.kiare-feed= 4-20.
100.keead. |
1000.thousane, or jeih cheead. |
Breton.333
10.dec. |
20.ueguend. |
30.tregond= 3-10.
40.deu ueguend= 2-20.
50.hanter hand= half hundred.
60.tri ueguend= 3-20.
70.dec ha tri ueguend= 10 + 60.
80.piar ueguend= 4-20.
90.dec ha piar ueguend= 10 + 80.
100.cand. |
120.hueh ueguend= 6-20.
140.seih ueguend= 7-20.
160.eih ueguend= 8-20.
180.nau ueguend= 9-20.
200.deu gand= 2-100.
240.deuzec ueguend= 12-20.
280.piarzec ueguend= 14-20.
300.tri hand, or pembzec ueguend. |
400.piar hand= 4-100.
1000.mil. |
These lists show that the native development of
the Celtic number systems, originally showing a strong
preference for the vigesimal method of progression, has
been greatly modified by intercourse with Teutonic
and Latin races. The higher numerals in all these
languages, and in Irish many of the lower also, are
seen at a glance to be decimal. Among the scales here
given the Breton, the legitimate descendant of the
ancient Gallic, is especially interesting; but here, just
as in the other Celtic tongues, when we reach 1000,
the familiar Latin term for that number appears in the
various corruptions of
mille, 1000, which was carried
into the Celtic countries by missionary and military
influences.
In connection with the Celtic language, mention
must be made of the persistent vigesimal element
which has held its place in French. The ancient
Gauls, while adopting the language of their conquerors,
so far modified the decimal system of Latin as to
replace the natural septante, 70, octante, 80, nonante,
90, by soixante-dix, 60-10,
quatre-vingt, 4-20, and quatrevingt-dix,
4-20-10. From 61 to 99 the French method
of counting is wholly vigesimal, except for the presence
of the one word soixante. In old French this element
was still more pronounced. Soixante had not
yet appeared; and 60 and 70 were treis vinz,
3-20, and
treis vinz et dis, 3-20 and 10 respectively. Also, 120
was six vinz, 6-20, 140 was sept-vinz, etc.334 How far
this method ever extended in the French language
proper, it is, perhaps, impossible to say; but from the
name of an almshouse, les quinze-vingts,335 which formerly
existed in Paris, and was designed as a home for 300
blind persons, and from the pembzek-ueguent,
15-20, of
the Breton, which still survives, we may infer that it
was far enough to make it the current system of
common life.
Europe yields one other example of vigesimal counting,
in the number system of the Basques. Like most
of the Celtic scales, the Basque seems to become decimal
above 100. It does not appear to be related to
any other European system, but to be quite isolated
philologically. The higher units, as mila, 1000, are
probably borrowed, and not native. The tens in the
Basque scale are:336
10.hamar. |
20.hogei. |
30.hogei eta hamar= 20 + 10.
40.berrogei= 2-20.
50.berrogei eta hamar= 2-20 + 10.
60.hirurogei= 3-20.
70.hirurogei eta hamar= 3-20 + 10.
80.laurogei= 4-20.
90.laurogei eta hamar= 4-20 + 10.
100.ehun. |
1000.milla. |
Besides these we find two or three numeral scales in
Europe which contain distinct traces of vigesimal counting,
though the scales are, as a whole, decidedly decimal.
The Danish, one of the essentially Germanic
languages, contains the following numerals:
30.tredive= 3-10.
40.fyrretyve= 4-10.
50.halvtredsindstyve= half (of 20) from 3-20.
60.tresindstyve= 3-20.
70.halvfierdsindstyve= half from 4-20.
80.fiirsindstyve= 4-20.
90.halvfemsindstyve= half from 5-20.
100.hundrede. |
Germanic number systems are, as a rule, pure decimal
systems; and the Danish exception is quite remarkable.
We have, to be sure, such expressions in English as
three score,
four score, etc., and the Swedish, Icelandic,
and other languages of this group have similar terms.
Still, these are not pure numerals, but auxiliary words
rather, which belong to the same category as
pair,
dozen,
dizaine, etc., while the Danish words just given
are the ordinary numerals which form a part of the
every-day vocabulary of that language. The method
by which this scale expresses 50, 70, and 90 is especially
noticeable. It will be met with again, and
further examples of its occurrence given.
In Albania there exists one single fragment of vigesimal
numeration, which is probably an accidental compound
rather than the remnant of a former vigesimal
number system. With this single exception the Albanian
scale is of regular decimal formation. A few of
the numerals are given for the sake of comparison:337
30.tridgiete= 3-10.
40.dizet= 2-20.
50.pesedgiete= 5-10.
60.giastedgiete= 6-10, etc.
Among the almost countless dialects of Africa we find
a comparatively small number of vigesimal number systems.
The powers of the negro tribes are not strongly
developed in counting, and wherever their numeral scales
have been taken down by explorers they have almost
always been found to be decimal or quinary-decimal.
The small number I have been able to collect are here
given. They are somewhat fragmentary, but are as
complete as it was possible to make them.
Affadeh.338
10.dekang. |
20.degumm. |
30.piaske. |
40.tikkumgassih= 20 × 2.
50.tikkumgassigokang= 20 × 2 + 10.
60.tikkumgakro= 20 × 3.
70.dungokrogokang= 20 × 3 + 10.
80.dukumgade= 20 × 4.
90.dukumgadegokang= 20 × 4 + 10.
100.miah (borrowed from the Arabs). |
Ibo.339
10.iri. |
20.ogu. |
30.ogu n-iri= 20 + 10,
or iri ato= 10 × 3. |
40.ogu abuo= 20 × 2,
or iri anno= 10 × 4. |
100.ogu ise= 20 × 5.
Vei.340
10.tan. |
20.mo bande= a person finished.
30.mo bande ako tan= 20 + 10.
40.mo fera bande= 2 × 20.
100.mo soru bande= 5 persons finished.
Yoruba.341
10.duup. |
20.ogu. |
30.ogbo. |
40.ogo-dzi= 20 × 2.
60.ogo-ta= 20 × 3.
80.ogo-ri= 20 × 4.
100.ogo-ru= 20 × 5.
120.ogo-fa= 20 × 6.
140.ogo-dze= 20 × 7.
160.ogo-dzo= 20 × 8, etc.
Efik.342
10.duup. |
20.edip. |
30.edip-ye-duup= 20 + 10.
40.aba= 20 × 2.
60.ata= 20 × 3.
80.anan= 20 × 4.
100.ikie. |
The Yoruba scale, to which reference has already been
made,
p. 70, again shows its peculiar structure, by continuing
its vigesimal formation past 100 with no interruption
in its method of numeral building. It will be
remembered that none of the European scales showed
this persistency, but passed at that point into decimal
numeration. This will often be found to be the case;
but now and then a scale will come to our notice whose
vigesimal structure is continued, without any break, on
into the hundreds and sometimes into the thousands.
Bongo.343
10.kih. |
20.mbaba kotu= 20 × 1.
40.mbaba gnorr= 20 × 2.
100.mbaba mui= 20 × 5.
Mende.344
10.pu. |
20.nu yela gboyongo mai= a man finished.
30.nu yela gboyongo mahu pu= 20 + 10.
40.nu fele gboyongo= 2 men finished.
100.nu lolu gboyongo= 5 men finished.
Nupe.345
10.gu-wo. |
20.esin. |
30.gbonwo. |
40.si-ba= 2 × 20.
50.arota. |
60.sita= 3 × 20.
70.adoni. |
80.sini= 4 × 20.
90.sini be-guwo= 80 + 10.
100.sisun= 5 × 20.
Logone.346
10.chkan. |
20.tkam. |
30.tkam ka chkan= 20 + 10.
40.tkam ksde= 20 × 2.
50.tkam ksde ka chkan= 40 + 10.
60.tkam gachkir= 20 × 3.
100.mia (from Arabic). |
1000.debu. |
Mundo.347
10.nujorquoi. |
20.tiki bere. |
30.tiki bire nujorquoi= 20 + 10.
40.tiki borsa= 20 × 2.
50.tike borsa nujorquoi= 40 + 10.
Mandingo.348
10.tang. |
20.mulu. |
30.mulu nintang= 20 + 10.
40.mulu foola= 20 × 2.
50.mulu foola nintang= 40 + 10.
60.mulu sabba= 20 × 3.
70.mulu sabba nintang= 60 + 10.
80.mulu nani= 20 × 4.
90.mulu nani nintang= 80 + 10.
100.kemi. |
This completes the scanty list of African vigesimal
number systems that a patient and somewhat extended
search has yielded. It is remarkable that the number is
no greater. Quinary counting is not uncommon in the
“Dark Continent,” and there is no apparent reason why
vigesimal reckoning should be any less common than
quinary. Any one investigating African modes of counting
with the material at present accessible, will find
himself hampered by the fact that few explorers have
collected any except the first ten numerals. This leaves
the formation of higher terms entirely unknown, and
shows nothing beyond the quinary or non-quinary character
of the system. Still, among those which Stanley,
Schweinfurth, Salt, and others have collected, by far the
greatest number are decimal. As our knowledge of
African languages is extended, new examples of the
vigesimal method may be brought to light. But our
present information leads us to believe that they will
be few in number.
In Asia the vigesimal system is to be found with
greater frequency than in Europe or Africa, but it is
still the exception. As Asiatic languages are much
better known than African, it is probable that the future
will add but little to our stock of knowledge on this
point. New instances of counting by twenties may still
be found in northern Siberia, where much ethnological
work yet remains to be done, and where a tendency
toward this form of numeration has been observed to
exist. But the total number of Asiatic vigesimal scales
must always remain small—quite insignificant in comparison
with those of decimal formation.
In the Caucasus region a group of languages is found,
in which all but three or four contain vigesimal systems.
These systems are as follows:
Abkhasia.349
10.zpha-ba. |
20.gphozpha= 2 × 10.
30.gphozphei zphaba= 20 + 10.
40.gphin-gphozpha= 2 × 20.
60.chin-gphozpha= 3 × 20.
80.phsin-gphozpha= 4 × 20.
100.sphki. |
Avari
10.antsh-go. |
20.qo-go. |
30.lebergo. |
40.khi-qogo= 2 × 20.
50.khiqojalda antshgo= 40 + 10.
60.lab-qogo= 3 × 20.
70.labqojalda antshgo= 60 + 10.
80.un-qogo= 4 × 20.
100.nusgo. |
Kuri
10.tshud. |
20.chad. |
30.channi tshud= 20 + 10.
40.jachtshur. |
50.jachtshurni tshud= 40 + 10.
60.put chad= 3 × 20.
70.putchanni tshud= 60 + 10.
80.kud-chad= 4 × 20.
90.kudchanni tshud= 80 + 10.
100.wis. |
Udi
10.witsh. |
20.qa. |
30.sa-qo-witsh= 20 + 10.
40.pha-qo= 2 × 20.
50.pha-qo-witsh= 40 + 10.
60.chib-qo= 3 × 20.
70.chib-qo-witsh= 60 + 10.
80.bip-qo= 4 × 20.
90.bip-qo-witsh= 80 + 10.
100.bats. |
1000.hazar (Persian). |
Tchetchnia
10.ith. |
20.tqa. |
30.tqe ith= 20 + 10.
40.sauz-tqa= 2 × 20.
50.sauz-tqe ith= 40 + 10.
60.chuz-tqa= 3 × 20.
70.chuz-tqe ith= 60 + 10.
80.w-iez-tqa= 4 × 20.
90.w-iez-tqe ith= 80 + 10.
100.b'e. |
1000.ezir (akin to Persian). |
Thusch
10.itt. |
20.tqa. |
30.tqa-itt= 20 + 10.
40.sauz-tq= 2 × 20.
50.sauz-tqa-itt= 40 + 10.
60.chouz-tq= 3 × 20.
70.chouz-tqa-itt= 60 + 10.
80.dhewuz-tq= 4 × 20.
90.dhewuz-tqa-itt= 80 + 10.
100.phchauz-tq= 5 × 20.
200.itsha-tq= 10 × 20.
300.phehiitsha-tq= 15 × 20.
1000.satsh tqauz-tqa itshatqa= 2 × 20 × 20 + 200.
Georgia
10.athi. |
20.otsi. |
30.ots da athi= 20 + 10.
40.or-m-otsi= 2 × 20.
50.ormots da athi= 40 + 10.
60.sam-otsi= 3 × 20.
70.samots da athi= 60 + 10.
80.othch-m-otsi= 4 × 20.
90.othmots da athi= 80 + 10.
100.asi. |
1000.ath-asi= 10 × 100.
Lazi
10.wit. |
20.öts. |
30.öts do wit= 20 × 10.
40.dzur en öts= 2 × 20.
50.dzur en öts do wit= 40 + 10.
60.dzum en öts= 3 × 20.
70.dzum en öts do wit= 60 + 10.
80.otch-an-öts= 4 × 20.
100.os. |
1000.silia (akin to Greek). |
Chunsag.350
10.ants-go. |
20.chogo. |
30.chogela antsgo= 20 + 10.
40.kichogo= 2 × 20.
50.kichelda antsgo= 40 + 10.
60.taw chago= 3 × 20.
70.taw chogelda antsgo= 60 + 10.
80.uch' chogo= 4 × 20.
90.uch' chogelda antsgo. |
100.nusgo. |
1000.asargo (akin to Persian). |
Dido.351
10.zino. |
20.ku. |
30.kunozino. |
40.kaeno ku= 2 × 20.
50.kaeno kuno zino= 40 + 10.
60.sonno ku= 3 × 20.
70.sonno kuno zino= 60 + 10.
80.uino ku= 4 × 20.
90.uino huno zino= 80 + 10.
100.bischon. |
400.kaeno kuno zino= 40 × 10.
Akari
10.entzelgu. |
20.kobbeggu. |
30.lowergu. |
40.kokawu= 2 × 20.
50.kikaldanske= 40 + 10.
60.secikagu. |
70.kawalkaldansku= 3 × 20 + 10.
80.onkuku= 4 × 20.
90.onkordansku= 4 × 20 + 10.
100.nosku. |
1000.askergu (from Persian). |
Circassia
10.psche. |
20.to-tsch. |
30.totsch-era-pschirre= 20 + 10.
40.ptl'i-sch= 4 × 10.
50.ptl'isch-era-pschirre= 40 + 10.
60.chi-tsch= 6 × 10.
70.chitsch-era-pschirre= 60 + 10.
80.toshitl= 20 × 4?
90.toshitl-era-pschirre= 80 + 10.
100.scheh. |
1000.min (Tartar) or schi-psche= 100 × 10.
The last of these scales is an unusual combination of
decimal and vigesimal. In the even tens it is quite
regularly decimal, unless 80 is of the structure suggested
above. On the other hand, the odd tens are
formed in the ordinary vigesimal manner. The reason
for this anomaly is not obvious. I know of no other
number system that presents the same peculiarity, and
cannot give any hypothesis which will satisfactorily
account for its presence here. In nearly all the examples
given the decimal becomes the leading element
in the formation of all units above 100, just as was
the case in the Celtic scales already noticed.
Among the northern tribes of Siberia the numeral
scales appear to be ruder and less simple than those
just examined, and the counting to be more consistently
vigesimal than in any scale we have thus far met
with. The two following examples are exceedingly interesting,
as being among the best illustrations of counting
by twenties that are to be found anywhere in the
Old World.
Tschukschi.352
10.migitken= both hands.
20.chlik-kin= a whole man.
30.chlikkin mingitkin parol= 20 + 10.
40.nirach chlikkin= 2 × 20.
100.milin chlikkin= 5 × 20.
200.mingit chlikkin= 10 × 20, i.e. 10 men.
1000.miligen chlin-chlikkin= 5 × 200, i.e. five (times) 10 men.
Aino.353
10.wambi. |
20.choz. |
30.wambi i-doehoz= 10 from 40.
40.tochoz= 2 × 20.
50.wambi i-richoz= 10 from 60.
60.rechoz= 3 × 20.
70.wambi [i?] inichoz= 10 from 80.
80.inichoz= 4 × 20.
90.wambi aschikinichoz= 10 from 100.
100.aschikinichoz= 5 × 20.
110.wambi juwanochoz= 10 from 120.
120.juwano choz= 6 × 20.
130.wambi aruwanochoz= 10 from 140.
140.aruwano choz= 7 × 20.
150.wambi tubischano choz= 10 from 160.
160.tubischano choz= 8 × 20.
170.wambi schnebischano choz= 10 from 180.
180.schnebischano choz= 9 × 20.
190.wambi schnewano choz= 10 from 200.
200.schnewano choz= 10 × 20.
300.aschikinichoz i gaschima chnewano choz= 5 × 20 + 10 × 20.
400.toschnewano choz= 2 × (10 × 20).
500.aschikinichoz i gaschima toschnewano choz= 100 + 400.
600.reschiniwano choz= 3 × 200.
700.aschikinichoz i gaschima reschiniwano choz= 100 + 600.
800.inischiniwano choz= 4 × 200.
900.aschikinichoz i gaschima inischiniwano choz= 100 + 800.
1000.aschikini schinewano choz= 5 × 200.
2000.wanu schinewano choz= 10 × (10 × 20).
This scale is in one sense wholly vigesimal, and in
another way it is not to be regarded as pure, but as
mixed. Below 20 it is quinary, and, however far it
might be extended, this quinary element would remain,
making the scale quinary-vigesimal. But in another
sense, also, the Aino system is not pure. In any unmixed
vigesimal scale the word for 400 must be a
simple word, and that number must be taken as the
vigesimal unit corresponding to 100 in the decimal
scale. But the Ainos have no simple numeral word
for any number above 20, forming all higher numbers
by combinations through one or more of the processes
of addition, subtraction, and multiplication. The only
number above 20 which is used as a unit is 200, which
is expressed merely as 10 twenties. Any even number
of hundreds, or any number of thousands, is then
indicated as being so many times 10 twenties; and
the odd hundreds are so many times 10 twenties, plus
5 twenties more. This scale is an excellent example
of the cumbersome methods used by uncivilized races
in extending their number systems beyond the ordinary
needs of daily life.
In Central Asia a single vigesimal scale comes to
light in the following fragment of the Leptscha scale,
of the Himalaya region:354
10.kati. |
40.kafali= 4 × 10,
or kha nat= 2 × 20. |
50.kafano= 5 × 10,
or kha nat sa kati= 2 × 20 + 10. |
100.gjo, or kat. |
Further to the south, among the Dravidian races, the
vigesimal element is also found. The following will
suffice to illustrate the number systems of these dialects,
which, as far as the material at hand shows, are
different from each other only in minor particulars:
Mundari.355
10.gelea. |
20.mi hisi. |
30.mi hisi gelea= 20 + 10.
40.bar hisi= 2 × 20.
60.api hisi= 3 × 20.
80.upun hisi= 4 × 20.
100.mone hisi= 5 × 20.
In the Nicobar Islands of the Indian Ocean a well-developed
example of vigesimal numeration is found.
The inhabitants of these islands are so low in the scale
of civilization that a definite numeral system of any
kind is a source of some surprise. Their neighbours,
the Andaman Islanders, it will be remembered, have
but two numerals at their command; their intelligence
does not seem in any way inferior to that of the Nicobar
tribes, and one is at a loss to account for the
superior development of the number sense in the case
of the latter. The intercourse of the coast tribes with
traders might furnish an explanation of the difficulty
were it not for the fact that the numeration of the inland
tribes is quite as well developed as that of the
coast tribes; and as the former never come in contact
with traders and never engage in barter of any kind
except in the most limited way, the conclusion seems
inevitable that this is merely one of the phenomena of
mental development among savage races for which we
have at present no adequate explanation. The principal
numerals of the inland and of the coast tribes are:356
Inland Tribes
10.teya. |
20.heng-inai. |
30.heng-inai-tain= 20 + 5 (couples).
40.au-inai= 2 × 20.
100.tain-inai= 5 × 20.
200.teya-inai= 10 × 20.
300.teya-tain-inai= (10 + 5) × 20.
400.heng-teo. |
Coast Tribes
10.sham. |
20.heang-inai. |
30.heang-inai-tanai= 20 + 5 (couples).
40.an-inai= 2 × 20.
100.tanai-inai= 5 × 20.
200.sham-inai= 10 × 20.
300.heang-tanai-inai= (10 + 5) 20.
400.heang-momchiama. |
In no other part of the world is vigesimal counting
found so perfectly developed, and, among native races,
so generally preferred, as in North and South America.
In the eastern portions of North America and in
the extreme western portions of South America the
decimal or the quinary decimal scale is in general
use. But in the northern regions of North America, in
western Canada and northwestern United States, in
Mexico and Central America, and in the northern and
western parts of South America, the unit of counting
among the great majority of the native races was 20.
The ethnological affinities of these races are not yet
definitely ascertained; and it is no part of the scope of
this work to enter into any discussion of that involved
question. But either through contact or affinity, this
form of numeration spread in prehistoric times over
half or more than half of the western hemisphere. It
was the method employed by the rude Eskimos of the
north and their equally rude kinsmen of Paraguay and
eastern Brazil; by the forest Indians of Oregon and
British Columbia, and by their more southern kinsmen,
the wild tribes of the Rio Grande and of the Orinoco.
And, most striking and interesting of all, it was the
method upon which were based the numeral systems of
the highly civilized races of Mexico, Yucatan, and New
Granada. Some of the systems obtained from the languages
of these peoples are perfect, extended examples
of vigesimal counting, not to be duplicated in any
other quarter of the globe. The ordinary unit was, as
would be expected, “one man,” and in numerous languages
the words for 20 and man are identical. But
in other cases the original meaning of that numeral
word has been lost; and in others still it has a signification
quite remote from that given above. These
meanings will be noticed in connection with the scales
themselves, which are given, roughly speaking, in their
geographical order, beginning with the Eskimo of the
far north. The systems of some of the tribes are as
follows:
Alaskan Eskimos.357
10.koleet. |
20.enuenok. |
30.enuenok kolinik= 20 + 10.
40.malho kepe ak= 2 × 20.
50.malho-kepe ak-kolmik che pah ak to= 2 × 20 + 10.
60.pingi shu-kepe ak= 3 × 20.
100.tale ma-kepe ak= 5 × 20.
400.enue nok ke pe ak= 20 × 20.
Tchiglit.358
10.krolit. |
20.kroleti, or innun= man.
30.innok krolinik-tchikpalik= man + 2 hands.
40.innum mallerok= 2 men.
50.adjigaynarmitoat= as many times 10 as the fingers of the hand.
60.innumipit= 3 men.
70.innunmalloeronik arveneloerit= 7 men?
80.innun pinatçunik arveneloerit= 8 men?
90.innun tcitamanik arveneloerit= 9 men?
100.itchangnerkr. |
1000.itchangner-park= great 100.
The meanings for 70, 80, 90, are not given by Father
Petitot, but are of such a form that the significations
seem to be what are given above. Only a full acquaintance
with the Tchiglit language would justify one in
giving definite meanings to these words, or in asserting
that an error had been made in the numerals. But it
is so remarkable and anomalous to find the decimal
and vigesimal scales mingled in this manner that one
involuntarily suspects either incompleteness of form, or
an actual mistake.
Tlingit.359
10.djinkat= both hands?
20.tle ka= 1 man.
30.natsk djinkat= 3 × 10.
40.dak'on djinkat= 4 × 10.
50.kedjin djinkat= 5 × 10.
60.tle durcu djinkat= 6 × 10.
70.daqa durcu djinkat= 7 × 10.
80.natska durcu djinkat= 8 × 10.
90.gocuk durcu djinkat= 9 × 10.
100.kedjin ka= 5 men, or 5 × 20.
200.djinkat ka= 10 × 20.
300.natsk djinkat ka= 30 men.
400.dak'on djinkat ka= 40 men.
This scale contains a strange commingling of decimal
and vigesimal counting. The words for 20, 100, and
200 are clear evidence of vigesimal, while 30 to 90, and
the remaining hundreds, are equally unmistakable proof
of decimal, numeration. The word
ka, man, seems to
mean either 10 or 20; a most unusual occurrence.
The fact that a number system is partly decimal and
partly vigesimal is found to be of such frequent occurrence
that this point in the Tlingit scale need excite
no special wonder. But it is remarkable that the same
word should enter into numeral composition under such
different meanings.
Nootka.360
10.haiu. |
20.tsakeits. |
30.tsakeits ic haiu= 20 + 10.
40.atlek= 2 × 20.
60.katstsek= 3 × 20.
80.moyek= 4 × 20.
100.sutc'ek= 5 × 20.
120.nop'ok= 6 × 20.
140.atlpok= 7 × 20.
160.atlakutlek= 8 × 20.
180.ts'owakutlek= 9 × 20.
200.haiuk= 10 × 20.
This scale is quinary-vigesimal, with no apparent
decimal element in its composition. But the derivation
of some of the terms used is detected with difficulty.
In the following scale the vigesimal structure is still
more obscure.
Tsimshian.361
10.gy'ap. |
20.kyedeel= 1 man.
30.gulewulgy'ap. |
40.t'epqadalgyitk, or tqalpqwulgyap. |
50.kctoncwulgyap. |
100.kcenecal. |
200.k'pal. |
300.k'pal te kcenecal= 200 + 100.
400.kyedal. |
500.kyedal te kcenecal= 400 + 100.
600.gulalegyitk. |
700.gulalegyitk te kcenecal= 600 + 100.
800.tqalpqtalegyitk. |
900.tqalpqtalegyitk te kcenecal= 800 + 100.
1000.k'pal. |
To the unobservant eye this scale would certainly
appear to contain no more than a trace of the vigesimal
in its structure. But Dr. Boas, who is one of
the most careful and accurate of investigators, says in
his comment on this system: “It will be seen at once
that this system is quinary-vigesimal.… In 20 we
find the word gyat, man. The hundreds are identical
with the numerals used in counting men (see p. 87),
and then the quinary-vigesimal system is most evident.”
Rio Norte Indians.362
20.taiguaco. |
30.taiguaco co juyopamauj ajte= 20 + 2 × 5.
40.taiguaco ajte= 20 × 2.
50.taiguaco ajte co juyopamauj ajte= 20 × 2 + 5 × 2.
Caribs of Essiquibo, Guiana
10.oween-abatoro. |
20.owee-carena= 1 person.
40.oko-carena= 2 persons.
60.oroowa-carena= 3 persons.
Otomi
10.ra-tta. |
20.na-te. |
30.na-te-m'a-ratta= 20 + 10.
40.yo-te= 2 × 30.
50.yote-m'a-ratta= 2 × 20 + 10.
60.hiu-te= 3 × 20.
70.hiute-m'a-ratta= 3 × 20 + 10.
80.gooho-rate= 4 × 20.
90.gooho-rate-m'a ratta= 4 × 20 + 10.
100.cytta-te= 5 × 20,
or nanthebe= 1 × 100. |
Maya, Yucatan.363
1.hun. |
10.lahun= it is finished.
20.hunkal= a measure, or more correctly, a fastening together.
30.lahucakal= 40 − 10?
40.cakal= 2 × 20.
50.lahuyoxkal= 60 − 10.
60.oxkal= 3 × 20.
70.lahucankal= 80 − 10.
80.cankal= 4 × 20.
90.lahuyokal= 100 − 10.
100.hokal= 5 × 20.
110.lahu uackal= 120 − 10.
120.uackal= 6 × 20.
130.lahu uuckal= 140 − 10.
140.uuckal= 7 × 20.
200.lahuncal= 10 × 20.
300.holhukal= 15 × 20.
400.hunbak= 1 tying around.
500.hotubak. |
600.lahutubak |
800.calbak= 2 × 400.
900.hotu yoxbak. |
1000.lahuyoxbak. |
1200.oxbak= 3 × 400.
2000.capic (modern). |
8000.hunpic= 1 sack.
16,000.ca pic (ancient). |
160,000.calab= a filling full
3,200,000.kinchil. |
64,000,000.hunalau. |
In the Maya scale we have one of the best and most
extended examples of vigesimal numeration ever developed
by any race. To show in a more striking and forcible
manner the perfect regularity of the system, the
following tabulation is made of the various Maya units,
which will correspond to the “10 units make one ten,
10 tens make one hundred, 10 hundreds make one thousand,”
etc., which old-fashioned arithmetic compelled us
to learn in childhood. The scale is just as regular by
twenties in Maya as by tens in English. It is
364
20 hun= 1 kal= 20.
20 kal= 1 bak= 400.
20 bak= 1 pic= 8000.
20 pic= 1 calab= 160,000.
20 calab= 1{kinchil | } | tzotzceh= 3,200,000. |
20 kinchil= 1 alau= 64,000,000.
The original meaning of pic, given in the scale as
“a sack,” was rather “a short petticoat, somtimes used
as a sack.” The word tzotzceh signified “deerskin.”
No reason can be given for the choice of this word as
a numeral, though the appropriateness of the others is
sufficiently manifest. No evidence of digital numeration
appears in the first 10 units, but, judging from
the almost universal practice of the Indian tribes of
both North and South America, such may readily have
been the origin of Maya counting. Whatever its origin,
it certainly expanded and grew into a system whose
perfection challenges our admiration. It was worthy of
the splendid civilization of this unfortunate race, and,
through its simplicity and regularity, bears ample testimony
to the intellectual capacity which originated it.
The only example of vigesimal reckoning which is comparable
with that of the Mayas is the system employed
by their northern neighbours, the Nahuatl, or, as they are
more commonly designated, the Aztecs of Mexico. This
system is quite as pure and quite as simple as the Maya,
but differs from it in some important particulars. In
its first 20 numerals it is quinary (see p. 141), and as
a system must be regarded as quinary-vigesimal. The
Maya scale is decimal through its first 20 numerals,
and, if it is to be regarded as a mixed scale, must
be characterized as decimal-vigesimal. But in both
these instances the vigesimal element preponderates so
strongly that these, in common with their kindred number
systems of Mexico, Yucatan, and Central America,
are always thought of and alluded to as vigesimal
scales. On account of its importance, the Nahuatl system365
is given in fuller detail than most of the other
systems I have made use of.
10.matlactli= 2 hands.
20.cempoalli= 1 counting.
21.cempoalli once= 20-1.
22.cempoalli omome= 20-2.
30.cempoalli ommatlactli= 20-10.
31.cempoalli ommatlactli once= 20-10-1.
40.ompoalli= 2 × 20.
50.ompoalli ommatlactli= 40-10.
60.eipoalli, or epoalli,= 3 × 20.
70.epoalli ommatlactli= 60-10.
80.nauhpoalli= 4 × 20.
90.nauhpoalli ommatlactli= 9080-10.
100.macuilpoalli= 5 × 20.
120.chiquacempoalli= 6 × 20.
140.chicompoalli= 7 × 20.
160.chicuepoalli= 8 × 20.
180.chiconauhpoalli= 9 × 20.
200.matlacpoalli= 10 × 20.
220.matlactli oncempoalli= 11 × 20.
240.matlactli omompoalli= 12 × 20.
260.matlactli omeipoalli= 13 × 20.
280.matlactli onnauhpoalli= 14 × 20.
300.caxtolpoalli= 15 × 20.
320.caxtolli oncempoalli. |
399.caxtolli onnauhpoalli ipan caxtolli onnaui= 19 × 20 + 19.
400.centzontli= 1 bunch of grass, or 1 tuft of hair.
800.ometzontli= 2 × 400.
1200.eitzontli= 3 × 400.
7600.caxtolli onnauhtzontli= 19 × 400.
8000.cenxiquipilli, or cexiquipilli. |
160,000.cempoalxiquipilli= 20 × 8000.
3,200,000.centzonxiquipilli= 400 × 8000.
64,000,000.cempoaltzonxiquipilli= 20 × 400 × 8000.
Up to 160,000 the Nahuatl system is as simple and
regular in its construction as the English. But at this
point it fails in the formation of a new unit, or rather
in the expression of its new unit by a simple word;
and in the expression of all higher numbers it is forced
to resort in some measure to compound terms, just as
the English might have done had it not been able to
borrow from the Italian. The higher numeral terms,
under such conditions, rapidly become complex and cumbersome,
as the following analysis of the number 1,279,999,999
shows.366 The analysis will be readily understood
when it is remembered that ipan signifies plus. Caxtolli
onnauhpoaltzonxiquipilli ipan caxtolli onnauhtzonxiquipilli
ipan caxtolli onnauhpoalxiquipilli ipan caxtolli onnauhxiquipilli
ipan caxtolli onnauhtzontli ipan caxtolli onnauhpoalli
ipan caxtolli onnaui; i.e. 1,216,000,000 + 60,800,000
+ 3,040,000 + 152,000 + 7600 + 380 + 19. To show the
compounding which takes place in the higher numerals,
the analysis may be made more literally, thus:
(15 + 4) × 20 × 400 × 8000 + (15 + 4) × 400 × 8000 + (15 + 4) × 20 × 8000 + (15
+ 4) × 8000 + (15 + 4) × 400 + (15 + 4) × 20 + 15
+ 4. Of course this resolution suffers from the fact
that it is given in digits arranged in accordance with
decimal notation, while the Nahuatl numerals express
values by a base twice as great. This gives the effect
of a complexity and awkwardness greater than really
existed in the actual use of the scale. Except for the
presence of the quinary element the number just given
is really expressed with just as great simplicity as it
could be in English words if our words “million” and
“billion” were replaced by “thousand thousand” and
“thousand thousand thousand.” If Mexico had remained
undisturbed by Europeans, and science and commerce
had been left to their natural growth and development,
uncompounded words would undoubtedly have been
found for the higher units, 160,000, 3,200,000, etc.,
and the system thus rendered as simple as it is possible
for a quinary-vigesimal system to be.
Other number scales of this region are given as
follows:
Huasteca.367
10.laluh. |
20.hum-inic= 1 man.
30.hum-inic-lahu= 1 man 10.
40.tzab-inic= 2 men.
50.tzab-inic-lahu= 2 men 10.
60.ox-inic= 3 men.
70.ox-inic-lahu= 3 men 10.
80.tze-tnic= 4 men.
90.tze-ynic-kal-laluh= 4 men and 10.
100.bo-inic= 5 men.
200.tzab-bo-inic= 2 × 5 men.
300.ox-bo-inic= 3 × 5 men.
400.tsa-bo-inic= 4 × 5 men.
600.acac-bo-inic= 6 × 5 men.
800.huaxic-bo-inic= 8 × 5 men.
1000.xi. |
8000.huaxic-xi= 8-1000.
The essentially vigesimal character of this system
changes in the formation of some of the higher numerals,
and a suspicion of the decimal enters. One hundred is
boinic, 5 men; but 200, instead of being simply lahuh-inic,
10 men, is tsa-bo-inic, 2 × 100, or more strictly, 2 times
5 men. Similarly, 300 is 3 × 100, 400 is 4 × 100, etc.
The word for 1000 is simple instead of compound, and
the thousands appear to be formed wholly on the decimal
base. A comparison of this scale with that of the
Nahuatl shows how much inferior it is to the latter,
both in simplicity and consistency.
Totonaco.368
10.cauh. |
20.puxam. |
30.puxamacauh= 20 + 10.
40.tipuxam= 2 × 20.
50.tipuxamacauh= 40 + 10.
60.totonpuxam= 3 × 20.
100.quitziz puxum= 5 × 20.
200.copuxam= 10 × 20.
400.tontaman. |
1000.titamanacopuxam= 2 × 400 + 200.
The essential character of the vigesimal element is
shown by the last two numerals. Tontamen, the square
of 20, is a simple word, and 1000 is, as it should be,
2 times 400, plus 200. It is most unfortunate that the
numeral for 8000, the cube of 20, is not given.
Cora.369
10.tamoamata. |
20.cei-tevi. |
30.ceitevi apoan tamoamata= 20 + 10.
40.huapoa-tevi= 2 × 20.
60.huaeica-tevi= 3 × 20.
100.anxu-tevi= 5 × 20.
400.ceitevi-tevi= 20 × 20.
Closely allied with the Maya numerals and method
of counting are those of the Quiches of Guatemala. The
resemblance is so obvious that no detail in the Quiche
scale calls for special mention.
Quiche.370
10.lahuh. |
20.hu-uinac= 1 man.
30.hu-uinac-lahuh= 20 + 10.
40.ca-uinac= 2 men.
50.lahu-r-ox-kal= −10 + 3 × 20.
60.ox-kal= 3 × 20.
70.lahu-u-humuch= −10 + 80.
80.humuch. |
90.lahu-r-ho-kal= −10 + 100.
100.hokal. |
1000.o-tuc-rox-o-kal. |
Among South American vigesimal systems, the best
known is that of the Chibchas or Muyscas of the Bogota
region, which was obtained at an early date by the missionaries
who laboured among them. This system is
much less extensive than that of some of the more
northern races; but it is as extensive as almost any
other South American system with the exception of the
Peruvian, which was, however, a pure decimal system.
As has already been stated, the native races of South
America were, as a rule, exceedingly deficient in regard to
the number sense. Their scales are rude, and show great
poverty, both in formation of numeral words and in the
actual extent to which counting was carried. If extended
as far as 20, these scales are likely to become vigesimal,
but many stop far short of that limit, and no inconsiderable
number of them fail to reach even 5. In this
respect we are reminded of the Australian scales, which
were so rudimentary as really to preclude any proper
use of the word “system” in connection with them.
Counting among the South American tribes was often
equally limited, and even less regular. Following are
the significant numerals of the scale in question:
Chibcha, or Muysca.371
10.hubchibica. |
20.quihica ubchihica= thus says the foot, 10 = 10-10,
or gueta= house. |
30.guetas asaqui ubchihica= 20 + 10.
40.gue-bosa= 20 × 2.
60.gue-mica= 20 × 3.
80.gue-muyhica= 20 × 4.
100.gue-hisca= 20 × 5.
Nagranda.372
10.guha. |
20.dino. |
30.'badiñoguhanu= 20 + 10.
40.apudiño= 2 × 20.
50.apudiñoguhanu= 2 × 20 + 10.
60.asudiño= 3 × 20.
70.asudiñoguhanu= 3 × 20 + 10.
80.acudiño= 4 × 20.
90.acudiñoguhanu= 4 × 20 + 10.
100.huisudiño= 5 × 20,
or guhamba= great 10. |
200.guahadiño= 10 × 20.
400.diñoamba= great 20.
1000.guhaisudiño= 10 × 5 × 20.
2000.hisudiñoamba= 5 great 20's.
4000.guhadiñoamba= 10 great 20's.
In considering the influence on the manners and customs
of any people which could properly be ascribed
to the use among them of any other base than 10, it
must not be forgotten that no races, save those using
that base, have ever attained any great degree of civilization,
with the exception of the ancient Aztecs and
their immediate neighbours, north and south. For reasons
already pointed out, no highly civilized race has
ever used an exclusively quinary system; and all that
can be said of the influence of this mode of counting
is that it gives rise to the habit of collecting objects
in groups of five, rather than of ten, when any attempt
is being made to ascertain their sum. In the case of
the subsidiary base 12, for which the Teutonic races
have always shown such a fondness, the dozen and
gross of commerce, the divisions of English money, and
of our common weights and measures are probably an
outgrowth of this preference; and the Babylonian base,
60, has fastened upon the world forever a sexagesimal
method of dividing time, and of measuring the circumference
of the circle.
The advanced civilization attained by the races of
Mexico and Central America render it possible to see
some of the effects of vigesimal counting, just as a
single thought will show how our entire lives are influenced
by our habit of counting by tens. Among the
Aztecs the universal unit was 20. A load of cloaks, of
dresses, or other articles of convenient size, was 20.
Time was divided into periods of 20 days each. The
armies were numbered by divisions of 8000;373 and in
countless other ways the vigesimal element of numbers
entered into their lives, just as the decimal enters into
ours; and it is to be supposed that they found it as
useful and as convenient for all measuring purposes as
we find our own system; as the tradesman of to-day
finds the duodecimal system of commerce; or as the
Babylonians of old found that singularly curious system,
the sexagesimal. Habituation, the laws which the
habits and customs of every-day life impose upon us,
are so powerful, that our instinctive readiness to make
use of any concept depends, not on the intrinsic
perfection or imperfection which pertains to it, but
on the familiarity with which previous use has invested
it. Hence, while one race may use a decimal,
another a quinary-vigesimal, and another a sexagesimal
scale, and while one system may actually be inherently
superior to another, no user of one method of reckoning
need ever think of any other method as possessing
practical inconveniences, of which those employing it
are ever conscious. And, to cite a single instance
which illustrates the unconscious daily use of two
modes of reckoning in one scale, we have only to think
of the singular vigesimal fragment which remains to
this day imbedded in the numeral scale of the French.
In counting from 70 to 100, or in using any number
which lies between those limits, no Frenchman is conscious
of employing a method of numeration less simple
or less convenient in any particular, than when he
is at work with the strictly decimal portions of his
scale. He passes from the one style of counting to the
other, and from the second back to the first again,
entirely unconscious of any break or change; entirely
unconscious, in fact, that he is using any particular
system, except that which the daily habit of years has
made a part himself.
Deep regret must be felt by every student of philology,
that the primitive meanings of simple numerals
have been so generally lost. But, just as the pebble
on the beach has been worn and rounded by the beating
of the waves and by other pebbles, until no trace
of its original form is left, and until we can say of it
now only that it is quartz, or that it is diorite, so too
the numerals of many languages have suffered from the
attrition of the ages, until all semblance of their origin
has been lost, and we can say of them only that
they are numerals. Beyond a certain point we can
carry the study neither of number nor of number
words. At that point both the mathematician and the
philologist must pause, and leave everything beyond to
the speculations of those who delight in nothing else
so much as in pure theory.
The End.
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- Armstrong, R. A., 180.
- Aymonier, A., 156.
- Bachofen, J. J., 131.
- Balbi, A., 151.
- Bancroft, H. H., 29, 47, 89, 93, 113, 199.
- Barlow, H., 108.
- Beauregard, O., 45, 83, 152.
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- Burton, R. F., 37, 71.
- Chamberlain, A. F., 45, 65, 93.
- Chase, P. E., 99.
- Clarke, H., 113.
- Codrington, R. H., 16, 95, 96, 136, 138, 145, 153, 154.
- Crawfurd, J., 89, 93, 130.
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- Deschamps, M., 28.
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- Du Graty, A. M., 138.
- Ellis, A. A., 64, 91.
- Ellis, R., 37, 142.
- Ellis, W., 83, 119.
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- Flegel, R., 133.
- Gallatin, A., 136, 159, 166, 171, 199, 204, 206, 208.
- Galton, F., 4.
- Gatschet, A. S., 58, 59, 68.
- Gilij, F. S., 54.
- Gill, W. W., 18, 118.
- Goedel, M., 83, 147.
- Grimm, J. L. C., 48.
- Gröber, G., 182.
- Guillome, J., 181.
- Haddon, A. C., 18, 105.
- Hale, H., 61, 65, 93, 114–116, 122, 130, 156, 163, 164, 171.
- Hankel, H., 137.
- Haumonté, J. D., 44.
- Hervas, L., 170.
- Humboldt, A. von, 32, 207.
- Hyades, M., 22.
- Kelly, J. W., 157, 196.
- Kelly, J., 180.
- Kleinschmidt, S., 52, 80.
- Lang, J. D., 108.
- Lappenberg, J. M., 127.
- Latham, R. G., 24, 67, 107.
- Leibnitz, G. W. von, 102, 103.
- Lloyd, H. E., 7.
- Long, C. C., 148, 186.
- Long, S. H., 121.
- Lubbock, Sir J., 3, 5.
- Lull, E. P., 79.
- Macdonald, J., 15.
- Mackenzie, A., 26.
- Man, E. H., 28, 194.
- Mann, A., 47.
- Marcoy, P. (Saint Cricq), 23, 168.
- Mariner, A., 85.
- Martius, C. F. von, 23, 79, 111, 122, 138, 142, 174.
- Mason, 112.
- Mill, J. S., 3.
- Moncelon, M., 142.
- Morice, A., 15, 86.
- Müller, Fr., 10, 27, 28, 45, 48, 55, 56, 60, 63, 66, 69, 78, 80, 90, 108, 111, 121, 122, 130, 136, 139, 146–151, 156–158, 165–167, 185–187, 191, 193.
- Murdoch, J., 30, 49,137.
- Nystron, J. W., 132.
- O'Donovan, J., 180.
- Oldfield, A., 29, 77.
- Olmos, A. de, 141.
- Parisot, J., 44.
- Park, M., 145–147.
- Parry, W. E., 32.
- Peacock, G., 8, 56, 84, 111, 118, 119, 154, 186.
- Petitot, E., 53, 157, 196.
- Pott, A. F., 50, 68, 92, 120, 145, 148, 149, 152, 157, 166, 182, 184, 189, 191, 205.
- Pruner-Bey, 10, 104.
- Pughe, W. O., 141.
- Ralph, J., 125.
- Ray, S. H., 45, 78, 80.
- Ridley, W., 57.
- Roth, H. L., 79.
- Salt, H., 187.
- Sayce, A. H., 75.
- Schoolcraft, H. R., 66, 81, 83, 84, 159, 160.
- Schröder, P., 90.
- Schweinfurth, G., 143, 146, 149, 186, 187.
- Simeon, R., 201.
- Spix, J. B. von, 7.
- Spurrell, W., 180.
- Squier, G. E., 80, 207.
- Stanley, H. M., 38, 42, 64, 69, 78, 150, 187.
- Taplin, G., 106.
- Thiel, B. A., 172.
- Toy, C. H., 70.
- Turner, G., 152, 154.
- Tylor, E. B., 2, 3, 15, 18, 22, 63, 65, 78, 79, 81, 84, 97, 124.
- Van Eys, J. W., 182.
- Vignoli, T., 95.
- Wallace, A. R., 174.
- Wells, E. R., jr., 157, 196.
- Whewell, W., 3.
- Wickersham, J., 96.
- Wiener, C., 22.
- Williams, W. L., 123.
- Abacus, 19.
- Abeokuta, 33.
- Abipone, 71, 72.
- Abkhasia, 188.
- Aboker, 148.
- Actuary, Life ins., 19.
- Adaize, 162.
- Addition, 19, 43, 46, 92.
- Adelaide, 108.
- Admiralty Islands, 45.
- Affadeh, 184.
- Africa (African), 9, 16, 28, 29, 32, 33, 38, 42, 47, 64, 66, 69, 78, 80, 91, 105, 120, 145, 170, 176, 184, 187.
- Aino (Ainu), 45, 191.
- Akra, 120.
- Akari, 190.
- Alaska, 157, 196.
- Albania, 184.
- Albert River, 26.
- Aleut, 157.
- Algonkin (Algonquin), 45, 92, 161.
- Amazon, 23.
- Ambrym, 136.
- American, 10, 16, 19, 98, 105.
- Andaman, 8, 15, 28, 31, 76, 174, 193.
- Aneitum, 154.
- Animal, 3, 6.
- Anthropological, 21.
- Apho, 133.
- Api, 80, 136, 155.
- Apinage, 111.
- Arab, 170.
- Arawak, 52–54, 135.
- Arctic, 29.
- Arikara, 46.
- Arithmetic, 1, 5, 30, 33, 73, 93.
- Aryan, 76, 128–130.
- Ashantee, 145.
- Asia (Asiatic), 28, 113, 131, 187.
- Assiniboine, 66, 92.
- Athapaskan,92.
- Atlantic, 126.
- Aurora, 155.
- Australia (Australian), 2, 6, 19, 22, 24–30, 57, 58, 71, 75, 76, 84, 103, 105, 106, 110, 112, 118, 173, 206.
- Avari, 188.
- Aymara, 166.
- Aztec, 63, 78, 83, 89, 93, 201, 207, 208.
- Babusessé, 38.
- Babylonian, 208.
- Bagrimma, 148.
- Bahnars, 15.
- Bakairi, 111.
- Balad, 67.
- Balenque, 150.
- Bambarese, 95.
- Banks Islands, 16, 96, 153.
- Barea, 151.
- Bargaining, 18, 19, 32.
- Bari, 136.
- Barre, 174.
- Basa, 146.
- Basque, 40, 182.
- Bellacoola, see Bilqula.
- Belyando River, 109.
- Bengal, Bay of, 28.
- Benuë, 133.
- Betoya, 57, 112, 135, 140.
- Bilqula, 46, 164.
- Binary, chap. v.
- Binin, 149.
- Bird-nesting, 5.
- Bisaye, 90.
- Bogota, 206.
- Bolan, 120.
- Bolivia, 2, 21.
- Bongo, 143, 186.
- Bonzé, 151.
- Bororo, 23.
- Botocudo, 22, 31, 48, 71.
- Bourke, 108.
- Boyne River, 24.
- Brazil, 2, 7, 30, 174, 195.
- Bretagne (Breton), 120, 129, 181, 182.
- British Columbia, 45, 46, 65, 86, 88, 89, 112, 113, 195.
- Bullom, 147.
- Bunch, 64.
- Burnett River, 112.
- Bushman, 28, 31.
- Butong, 93.
- Caddoe, 162.
- Cahuillo, 165.
- Calculating machine, 19.
- Campa, 22.
- Canada, 29, 53, 54, 86, 195.
- Canaque, 142, 144.
- Caraja, 23.
- Carib, 166, 167, 199.
- Carnarvon, 35, 36.
- Carrier, 86.
- Carthaginian, 179.
- Caucasus, 188.
- Cayriri (see Kiriri), 79.
- Cayubaba (Cayubabi), 84, 167.
- Celtic, 40, 169, 179, 181, 190.
- Cely, Mom, 9.
- Central America, 29, 69, 79, 121, 131, 195, 201, 208.
- Ceylon, 28.
- Chaco, 22.
- Champion Bay, 109.
- Charles XII., 132.
- Cheyenne, 62.
- Chibcha, 206.
- China (Chinese), 40, 131.
- Chippeway, 62, 159, 162.
- Chiquito, 2, 6, 21, 40, 71, 76.
- Choctaw, 65, 85, 162.
- Chunsag, 189.
- Circassia, 190.
- Cobeu, 174.
- Cochin China, 15.
- Columbian, 113.
- Comanche, 29, 83.
- Conibo, 23.
- Cooper's Creek, 108.
- Cora, 166.
- Cotoxo, 111.
- Cowrie, 64, 70, 71.
- Cree, 91.
- Crocker Island, 107.
- Crow, 3, 4, 92.
- Crusoe, Robinson, 7.
- Curetu, 111.
- Dahomey, 71.
- Dakota, 81, 91, 92.
- Danish, 30, 46, 129, 183.
- Darnley Islands, 24.
- Delaware, 91, 160.
- Demara, 4, 6.
- Déné, 86.
- Dido, 189.
- Dinka, 136, 147.
- Dippil, 107.
- Division, 19.
- Dravidian, 104, 193.
- Dual number, 75.
- Duluth, 34.
- Duodecimal, chap. v.
- Dutch, 129.
- Eaw, 24.
- Ebon, 152.
- Efik, 148, 185.
- Encabellada, 22.
- Encounter Bay, 108.
- Ende, 68, 152.
- English, 28, 38–44, 60, 81, 85, 89, 118, 123, 124, 129, 183, 200, 203, 208.
- Eromanga, 96, 136, 154.
- Eskimo, 16, 30, 31, 32, 36, 48, 51, 52, 54, 61, 64, 83, 137, 157, 159, 195, 196.
- Essequibo, 166.
- Europe (European), 27, 39, 168, 169, 179, 182, 183, 185, 204.
- Eye, 14, 97.
- Eyer's Sand Patch, 26.
- Ewe, 64, 91.
- Fall, 163.
- Fate, 138, 155.
- Fatuhiva, 130.
- Feloop, 145.
- Fernando Po, 150.
- Fiji, 96.
- Finger pantomime, 10, 23, 29, 67, 82.
- Fingoe, 33.
- Fist, 16, 59, 72.
- Flinder's River, 24.
- Flores, 68, 152.
- Forefinger, 12, 15, 16, 54, 61, 91, 113.
- Foulah, 147.
- Fourth finger, 18.
- Frazer's Island, 108.
- French, 40, 41, 124, 129, 181, 182, 209.
- Fuegan, 22.
- Gaelic, 180.
- Galibi, 138.
- Gaul, 169, 182.
- Georgia, 189.
- German, 38–43, 129, 183.
- Gesture, 18, 59.
- Gola, 151.
- Golo, 146.
- Gonn Station, 110.
- Goth, 169.
- Greek, 86, 129, 168, 169.
- Green Island, 45.
- Greenland, 29, 52, 80, 158.
- Guachi, 23, 31.
- Guarani, 55, 138.
- Guatemala, 205.
- Guato, 142.
- Guaycuru, 22.
- Gudang, 24.
- Haida, 112.
- Hawaii, 113, 114, 116, 117.
- Head, 71.
- Heap, 8, 9, 25, 70, 77, 100.
- Hebrew, 86, 89, 95.
- Heiltsuk, 65, 88, 163.
- Herero, 150.
- Hervey Islands, 118.
- Hidatsa, 80, 91.
- Hill End, 109.
- Himalaya, 193.
- Hottentot, 80, 92.
- Huasteca, 204.
- Hudson's Bay, 48, 61.
- Hun, 169.
- Hunt, Leigh, 33.
- Ibo, 185.
- Icelandic, 129, 183.
- Illinois, 91.
- Index finger, 11, 14.
- India, 96, 112, 131.
- Indian, 8, 10, 13, 16, 17, 19, 32, 36, 54, 55, 59, 62, 65, 66, 79, 80, 82, 83, 89, 90, 98, 105, 112, 171, 201.
- Indian Ocean, 63, 193.
- Indo-European, 76.
- Irish, 129, 180.
- Italian, 39, 80, 124, 129, 203.
- Jajowerong, 156.
- Jallonkas, 146.
- Jaloff, 146.
- Japanese, 40, 86, 89, 93–95.
- Java, 93, 120.
- Jiviro, 61, 136.
- Joints of fingers, 7, 18, 173.
- Juri, 79.
- Kamassin, 130.
- Kamilaroi, 27, 107, 112.
- Kamtschatka, 75, 157.
- Kanuri, 136, 149.
- Karankawa, 68.
- Karen, 112.
- Keppel Bay, 24.
- Ki-Nyassa, 150.
- Kiriri, 55, 138, 139, 167.
- Kissi, 145.
- Ki-Swahili, 42.
- Ki-Yau, 150.
- Klamath, 58, 59.
- Knot, 7, 9, 19, 40, 93, 115.
- Kolyma, 75.
- Kootenay, 65.
- Koriak, 75.
- Kredy, 149.
- Kru, 146.
- Ku-Mbutti, 78.
- Kunama, 151.
- Kuri, 188.
- Kusaie, 78, 80.
- Kwakiutl, 45.
- Labillardière, 85.
- Labrador, 29.
- Lake Kopperamana, 107.
- Latin, 40, 44, 76, 81, 86, 124, 128, 168, 169, 181, 182.
- Lazi, 189.
- Left hand, 10–17, 54.
- Leper's Island, 16.
- Leptscha, 193.
- Lifu, 143.
- Little finger, 10–18, 48, 54, 61, 91.
- Logone, 186.
- London, 124.
- Lower California, 29.
- Luli, 118.
- Lutuami, 164.
- Maba, 80.
- Macassar, 93.
- Machine, Calculating, 19, 20.
- Mackenzie River, 157.
- Macuni, 174.
- Madagascar, 8, 9.
- Maipures, 15, 56.
- Mairassis, 10.
- Malagasy, 83, 95.
- Malanta, 96.
- Malay, 8, 45, 90, 93, 170.
- Mallicolo, 152.
- Manadu, 93.
- Mandingo, 186.
- Mangareva, 114.
- Manx, 180.
- Many, 2, 21–23, 25, 28, 100.
- Maori, 64, 93, 122.
- Marachowie, 26.
- Maré, 84.
- Maroura, 106.
- Marquesas, 93, 114, 115.
- Marshall Islands, 122, 152.
- Massachusetts, 91, 159.
- Mathematician, 2, 3, 35, 102, 127, 210.
- Matibani, 151.
- Matlaltzinca, 166.
- Maya, 45, 46, 199, 205.
- Mbayi, 111.
- Mbocobi, 22.
- Mbousha, 66.
- Melanesia, 16, 22, 28, 84, 95.
- Mende, 186.
- Mexico, 29, 195, 201, 204, 208.
- Miami, 91.
- Micmac, 90, 160.
- Middle finger, 12, 15, 62.
- Mille, 122.
- Minnal Yungar, 26.
- Minsi, 162.
- Mississaga, 44, 92.
- Mississippi, 125.
- Mocobi, 119.
- Mohegan, 91.
- Mohican, 172.
- Mokko, 149.
- Molele, 164.
- Moneroo, 109.
- Mongolian, 8.
- Montagnais, 53, 54, 175.
- Moree, 24.
- Moreton Bay, 108.
- Mort Noular, 107.
- Mosquito, 69, 70, 121.
- Mota, 95, 153.
- Mpovi, 152.
- Multiplication, 19, 33, 40, 43, 59.
- Mundari, 193.
- Mundo, 186.
- Muralug, 17.
- Murray River, 106, 109.
- Muysca, 206.
- Nagranda, 207.
- Nahuatl, 141, 144, 177, 201, 205.
- Nakuhiva, 116, 130.
- Negro, 8, 9, 15, 29, 184.
- Nengone, 63, 136.
- New, 128–130.
- New Caledonia, 154.
- New Granada, 195.
- New Guinea, 10, 152.
- New Hebrides, 155.
- New Ireland, 45.
- New Zealand, 123.
- Nez Perces, 65, 158.
- Ngarrimowro, 110.
- Niam Niam, 64, 136.
- Nicaragua, 80.
- Nicobar, 78, 193.
- Nightingale, 4.
- Nootka, 163, 198.
- Norman River, 24.
- North America, 28, 82, 171, 173, 176, 194, 201.
- Notch, 7, 9, 93.
- Numeral frame, 19.
- Nupe, 149, 186.
- Nusqually, 96.
- Oceania, 115, 176.
- Octonary, chap. v.
- Odessa, 34.
- Ojibwa, 84, 159.
- Okanaken, 88.
- Omaha, 161.
- Omeo, 110.
- Oregon, 58, 195.
- Orejone, 23.
- Orinoco, 54, 56, 195.
- Ostrich, 71, 72.
- Otomac, 15.
- Otomi, 165, 199.
- Ottawa, 159.
- Oyster Bay, 79.
- Pacific, 29, 113, 116, 117, 131.
- Palm (of the hand), 12, 14, 15.
- Palm Island, 156.
- Pama, 136, 155.
- Pampanaga, 66.
- Papaa, 148.
- Paraguay, 55, 71, 118, 195.
- Parana, 119.
- Paris, 182.
- Pawnee, 91, 121, 162.
- Pebble, 7–9, 19, 40, 93, 179.
- Peno, 2.
- Peru (Peruvian), 2, 22, 61, 206.
- Philippine, 66.
- Philology (Philologist), 128, 209, 210.
- Phœnician, 90, 179.
- Pigmy, 69, 70, 78.
- Pikumbul, 57, 138.
- Pines, Isle of, 153.
- Pinjarra, 26.
- Plenty, 25, 77.
- Point Barrow, 30, 51, 64, 83, 137, 159.
- Polynesia, 22, 28, 118, 130, 170.
- Pondo, 33.
- Popham Bay, 107.
- Port Darwin, 109.
- Port Essington, 24, 107.
- Port Mackay, 26.
- Port Macquarie, 109.
- Puget Sound, 96.
- Puri, 22, 92.
- Quappa, 171, 172.
- Quaternary, chap. v.
- Queanbeyan, 24.
- Quiche, 205.
- Quichua, 61.
- Rapid, 163.
- Rarotonga, 114.
- Richmond River, 109.
- Right hand, 10–18, 54.
- Right-handedness, 13, 14.
- Ring finger, 15.
- Rio Grande, 195.
- Rio Napo, 22.
- Rio Norte, 136, 199.
- Russia (Russian), 30, 35.
- Sahaptin, 158.
- San Antonio, 136.
- San Blas, 79, 80.
- Sanskrit, 40, 92, 97, 128.
- Sapibocone, 84, 167.
- Saste (Shasta), 165.
- Scratch, 7.
- Scythian, 169.
- Seed, 93.
- Semitic, 89.
- Senary, chap. v.
- Sesake, 136, 155.
- Several, 22.
- Sexagesimal, 124, 208.
- Shawnoe, 160.
- Shell, 7, 19, 70, 93.
- Shushwap, 88.
- Siberia, 29, 30, 187, 190.
- Sierra Leone, 83.
- Sign language, 6.
- Sioux, 83.
- Slang, 124.
- Slavonic, 40.
- Snowy River, 110.
- Soussou, 83, 147.
- South Africa, 4, 15, 28.
- South America, 2, 15, 22, 23, 27–29, 54, 57, 72, 76, 78, 79, 104, 110, 173, 174, 194, 201, 206.
- Spanish, 2, 23, 42.
- Splint, 7.
- Stick, 7, 179.
- Stlatlumh, 88.
- Streaky Bay, 26.
- String, 7, 9, 64, 71.
- Strong's Island, 78.
- Subtraction, 19, 44–47.
- Sunda, 120.
- Sweden (Swedish), 129, 132, 183.
- Tacona, 2.
- Taensa, 44.
- Tagala, 66.
- Tahiti, 114.
- Tahuata, 115.
- Tama, 111.
- Tamanac, 54, 135.
- Tambi, 120.
- Tanna, 154.
- Tarascan, 165.
- Tariana, 174.
- Tasmania, 24, 27, 79, 104, 106.
- Tawgy, 130.
- Tchetchnia, 188.
- Tchiglit, 157, 196.
- Tembu, 33.
- Temne, 148.
- Ternary, chap. v.
- Terraba, 172.
- Teutonic, 40, 41, 43, 179, 181, 208.
- Texas, 69.
- Thibet, 96.
- Thumb, 10–18, 54, 59, 61, 62, 113, 173.
- Thusch, 189.
- Ticuna, 168.
- Timukua, 165.
- Tlingit, 136, 163, 197.
- Tobi, 156.
- Tonga, 33, 85.
- Torres, 17, 96, 104, 105.
- Totonaco, 205.
- Towka, 78.
- Triton's Bay, 152.
- Tschukshi, 156, 191.
- Tsimshian, 86, 164, 198.
- Tweed River, 26.
- Uainuma, 122.
- Udi, 188.
- Uea, 67, 153.
- United States, 29, 83, 195.
- Upper Yarra, 110.
- Ureparapara, 153.
- Vaturana, 96.
- Vedda, 28, 31, 76, 174.
- Vei, 16, 147, 185.
- Victoria, 156.
- Vilelo, 60.
- Waiclatpu, 164.
- Wales (Welsh), 35, 46, 141, 144, 177, 180.
- Wallachia, 121.
- Warrego, 107, 109.
- Warrior Island, 107.
- Wasp, 5.
- Watchandie, 29, 77.
- Watji, 120.
- Weedookarry, 24.
- Wimmera, 107.
- Winnebago, 85.
- Wiraduroi, 27, 108.
- Wirri-Wirri, 108.
- Wokke, 112.
- Worcester, Mass., Schools of, 11.
- Yahua, 168.
- Yaruro, 139.
- Yengen, 154.
- Yit-tha, 109.
- Yoruba, 33, 47, 64, 70, 185.
- Yucatan, 195, 201.
- Yuckaburra, 26.
- Zamuco, 55, 60, 138, 139.
- Zapara, 111.
- Zulu, 16, 62.
- Zuñi, 13, 14, 48, 49, 53, 54, 60, 83, 137.
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