The Number Concept_ Its Origin and Development Levi L. Conant 828 downloads.pdf

THE PROJECT GUTENBERG EBOOK OF THE NUMBER
CONCEPT: ITS ORIGIN AND DEVELOPMENT

This eBook is for the use of anyone anywhere in the United States and most
other parts of the world at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms of
the Project Gutenberg License included with this eBook or online at
www.gutenberg.org. If you are not located in the United States, you will
have to check the laws of the country where you are located before using
this eBook.

Title: The Number Concept: Its Origin and Development

Author: Levi L. Conant

Release date: August 5, 2005 [eBook #16449]
Most recently updated: December 12, 2020

Language: English

Other information and formats: www.gutenberg.org/ebooks/16449

Credits: Produced by Jonathan Ingram, Hagen von Eitzen and the
Online Distributed Proofreading Team at https://www.pgdp.net

*** START OF THE PROJECT GUTENBERG EBOOK THE NUMBER
CONCEPT: ITS ORIGIN AND DEVELOPMENT ***

- ii - THE MACMILLAN COMPANY
NEW YORK · BOSTON · CHICAGO · DALLAS
ATLANTA · SAN FRANCISCO

MACMILLAN & CO., Limited
LONDON · BOMBAY · CALCUTTA
MELBOURNE

THE MACMILLAN COMPANY
OF CANADA, Limited
TORONTO

- iii - 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,
- iv -
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.

-v- PREFACE.

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 - vi -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.

- vii - CONTENTS.

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

THE NUMBER CONCEPT: ITS
-1-

ORIGIN AND DEVELOPMENT.

CHAPTER I.
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 - 2 -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 - 3 -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 - 4 -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 - 5 -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 - 6 -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 - 7 -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 - 8 -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 - 9 -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 -
10 -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 - 11 -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 - 12 -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 - 13 -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 - 14 -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 - 15 -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 - 16 -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 - 17 -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 - 18 -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, - 19 -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 - 20 -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.

- 21 - CHAPTER II.
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 - 22 -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 - 23 -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.
- 24 -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. - 25 -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 - 26 -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 - 27 -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
- 28 -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, - 29 -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 - 30 -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.
- 31 -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, - 32 -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 - 33 -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 - 34 -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 - 35 -
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?’ - 36 -‘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.

- 37 - CHAPTER III.
THE ORIGIN OF NUMBER WORDS.

In the comparison of languages and the search for primitive root forms,
no class of expressions has been subjected to closer scrutiny than the little
cluster of words, found in each language, which constitutes a part of the
daily vocabulary of almost every human being—the words with which we
begin our counting. It is assumed, and with good reason, that these are
among the earlier words to appear in any language; and in the mutations of
human speech, they are found to suffer less than almost any other portion of
a language. Kinship between tongues remote from each other has in many
instances been detected by the similarity found to exist among the every-
day words of each; and among these words one may look with a good
degree of certainty for the 1, 2, 3, etc., of the number scale. So fruitful has
been this line of research, that the attempt has been made, even, to establish
a common origin for all the races of mankind by means of a comparison of
numeral words.51 But in this instance, as in so many others that will readily
occur to the mind, the result has been that - 38 -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 - 39 -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, - 40 -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 - 41 -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 - 42 -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 - 43 -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. - 44 -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 - 45 -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 - 46 -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, - 47 -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 - 48 -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.

- 49 -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 - 50 -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 - 51 -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 - 52 -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.
- 53 -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.

{ inl'as dinri
t'a-ye-oyertan = there are still 3 of them.
7.
= 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 - 54 -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:

- 55 -5. amgnaitone = 1 hand complete.
6. itacono amgna pona tevinitpe = 1 on the other hand.
10. amgna aceponare = all of the 2 hands.
11. puitta pona tevinitpe = 1 on the foot.
16. itacono puitta pona tevinitpe = 1 on the other foot.
20. tevin itoto = 1 man.
21. itacono itoto jamgnar bona tevinitpe = 1 on the hands of another man.

In the Guarani75 language of Paraguay the same method is found, with a
different form of expression for 20. Here the numerals in question are

5. asepopetei = one hand.
10. asepomokoi = two hands.

20. asepo asepi abe = hands and feet.

Another slight variation is furnished by the Kiriri language,76 which is
also one of the numerous South American Indian forms of speech, where
we find the words to be

5. mi biche misa = one hand.
10. mikriba misa sai = both hands.
20. mikriba misa idecho ibi sai = both hands together with the feet.

Illustrations of this kind might be multiplied almost indefinitely; and it is
well to note that they may be drawn from all parts of the world. South
America is peculiarly rich in native numeral words of this kind; and, as the
examples above cited show, it is the field to which one instinctively turns
when this subject is under discussion. The Zamuco numerals are, among - 56 -
others, exceedingly interesting, giving us still a new variation in method.
They are77

1. tsomara.
2. gar.
3. gadiok.
4. gahagani.
5. tsuena yimana-ite = ended 1 hand.
6. tsomara-hi = 1 on the other.
7. gari-hi = 2 on the other.
8. gadiog-ihi = 3 on the other.
9. gahagani-hi = 4 on the other.
10. tsuena yimana-die = ended both hands.
11. tsomara yiri-tie = 1 on the foot.
12. gar yiritie = 2 on the foot.
20. tsuena yiri-die = ended both feet.

As is here indicated, the form of progression from 5 to 10, which we
should expect to be “hand-1,” or “hand-and-1,” or some kindred expression,
signifying that one hand had been completed, is simply “1 on the other.”

Again, the expressions for 11, 12, etc., are merely “1 on the foot,” “2 on the
foot,” etc., while 20 is “both feet ended.”
An equally interesting scale is furnished by the language of the
Maipures78 of the Orinoco, who count

1. papita.
2. avanume.
3. apekiva.
4. apekipaki.
5. papitaerri capiti = 1 only hand.
6. papita yana pauria capiti purena = 1 of the other hand we take.
10. apanumerri capiti = 2 hands.
11. papita yana kiti purena = 1 of the toes we take.
20. papita camonee = 1 man.
40. avanume camonee = 2 men.
60. apekiva camonee = 3 men, etc.

In all the examples thus far given, 20 is expressed either by the
- 57 -
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.
- 58 -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
- 59 -
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 - 60 -
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
- 61 -
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, - 62 -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 - 63 -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 - 64 -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.
- 65 -7. batiwwi = 2d 2.
8. batti-biata = 2d 3.
9. batti-biama = 2d 4.
10. bauwé = 2d 5.

That this method of progression is not confined to the least developed
languages, however, is shown by a most cursory examination of the
numerals of our American Indian tribes, where numeral formation like that
exhibited above is exceedingly common. In the Kootenay dialect,93 of
British Columbia, qaetsa, 4, and wo-qaetsa, 8, are obviously related, the
latter word probably meaning a second 4. Most of the native languages of
British Columbia form their words for 7 and 8 from those which signify 2

and 3; as, for example, the Heiltsuk,94 which shows in the following words
a most obvious correspondence:

2. matl. 7. matlaaus.
3. yutq. 8. yutquaus.

In the Choctaw language95 the relation between 2 and 7, and 3 and 8, is
no less clear. Here the words are:

2. tuklo. 7. untuklo.
3. tuchina. 8. untuchina.

The Nez Percés96 repeat the first three words of their scale in their 6, 7,
and 8 respectively, as a comparison of these numerals will show.

- 66 -1. naks. 6. oilaks.
2. lapit. 7. oinapt.
3. mitat. 8. oimatat.

In all these cases the essential point of the method is contained in the
repetition, in one way or another, of the numerals of the second quinate,
without the use with each one of the word for 5. This may make 6, 7, 8, and
9 appear as second 1, second 2, etc., or another 1, another 2, etc.; or, more
simply still, as 1 more, 2 more, etc. It is the method which was briefly
discussed in the early part of the present chapter, and is by no means
uncommon. In a decimal scale this repetition would begin with 11 instead
of 6; as in the system found in use in Tagala and Pampanaga, two of the
Philippine Islands, where, for example, 11, 12, and 13 are:97

11. labi-n-isa = over 1.
12. labi-n-dalaua = over 2.
13. labi-n-tatlo = over 3.

A precisely similar method of numeral building is used by some of our
Western Indian tribes. Selecting a few of the Assiniboine numerals98 as an
illustration, we have

11. ak kai washe = more 1.
12. ak kai noom pah = more 2.
13. ak kai yam me nee = more 3.
14. ak kai to pah = more 4.
15. ak kai zap tah = more 5.
16. ak kai shak pah = more 6, etc.

A still more primitive structure is shown in the numerals of the
Mboushas99 of Equatorial Africa. Instead of using 5-1, 5-2, 5-3, 5-4, or 2d
1, 2d 2, 2d 3, - 67 -2d 4, in forming their numerals from 6 to 9, they proceed
in the following remarkable and, at first thought, inexplicable manner to
form their compound numerals:

1. ivoco.
2. beba.
3. belalo.
4. benai.
5. betano.
6. ivoco beba = 1-2.
7. ivoco belalo = 1-3.
8. ivoco benai = 1-4.
9. ivoco betano = 1-5.
10. dioum.

No explanation is given by Mr. du Chaillu for such an apparently
incomprehensible form of expression as, for example, 1-3, for 7. Some
peculiar finger pantomime may accompany the counting, which, were it
known, would enlighten us on the Mbousha's method of arriving at so
anomalous a scale. Mere repetition in the second quinate of the words used
in the first might readily be explained by supposing the use of fingers
absolutely indispensable as an aid to counting, and that a certain word
would have one meaning when associated with a certain finger of the left

hand, and another meaning when associated with one of the fingers of the
right. Such scales are, if the following are correct, actually in existence
among the islands of the Pacific.

Balad.100 Uea.100
1. parai. 1. tahi.
2. paroo. 2. lua.
3. pargen. 3. tolu.
4. parbai. 4. fa.
5. panim. 5. lima.
6. parai. 6. tahi.
7. paroo. 7. lua.
8. pargen. 8. tolu.
9. parbai. 9. fa.
10. panim. 10. lima.

Such examples are, I believe, entirely unique among primitive number
- 68 -
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 - 69 -coast region of Texas. The first ten numerals of this
singular array are:

1. natsa.
2. haikia.
3. kachayi.
4. hayo hakn = 2 × 2.
5. natsa behema = 1 father, i.e. of the fingers.
6. hayo haikia = 3 × 2?
7. haikia natsa = 2 + 5?
8. haikia behema = 2 fathers?
9. haikia doatn = 2d from 10?
10. doatn habe.

Systems like the above, where chaos instead of order seems to be the
ruling principle, are of occasional occurrence, but they are decidedly the
exception.
In some of the cases that have been adduced for illustration it is to be
noticed that the process of combination begins with 7 instead of with 6.
Among others, the scale of the Pigmies of Central Africa103 and that of the
Mosquitos104 of Central America show this tendency. In the Pigmy scale the
words for 1 and 6 are so closely akin that one cannot resist the impression
that 6 was to them a new 1, and was thus named.

Mosquito. Pigmy.
1. kumi. ujju.
2. wal. ibari.
3. niupa. ikaro.
4. wal-wal = 2-2. ikwanganya.

Mosquito. Pigmy.
5. mata-sip = fingers of 1 hand. bumuti.
6. matlalkabe. ijju.
7. matlalkabe pura kumi = 6 and 1. bumutti-na-ibali = 5 and 2.
- 70 -8. matlalkabe pura wal = 6 and 2. bumutti-na-ikaro = 5 and 3.
9. matlalkabe pura niupa = 6 and 3. bumutti-na-ikwanganya = 5 and 4.
10. mata wal sip = fingers of 2 hands. mabo = half man.

The Mosquito scale is quite exceptional in forming 7, 8, and 9 from 6,
instead of from 5. The usual method, where combinations appear between 6
and 10, is exhibited by the Pigmy scale. Still another species of numeral
form, quite different from any that have already been noticed, is found in
the Yoruba105 scale, which is in many respects one of the most peculiar in
existence. Here the words for 11, 12, etc., are formed by adding the suffix -
la, great, to the words for 1, 2, etc., thus:

1. eni, or okan.
2. edzi.
3. eta.
4. erin.
5. arun.
6. efa.
7. edze.
8. edzo.
9. esan.
10. ewa.
11. okanla = great 1.
12. edzila = great 2.
13. etala = great 3.
14. erinla = great 4, etc.
40. ogodzi = string.
200. igba = heap.

The word for 40 was adopted because cowrie shells, which are used for
counting, were strung by forties; and igba, 200, because a heap of 200
shells was five strings, and thus formed a convenient higher unit for - 71 -

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 - 72 -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 - 73 -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.

- 74 - CHAPTER IV.
THE ORIGIN OF NUMBER WORDS.
(Continued.)

By the slow, and often painful, process incident to the extension and
development of any mental conception in a mind wholly unused to
abstractions, the savage gropes his way onward in his counting from 1, or
more probably from 2, to the various higher numbers required to form his
scale. The perception of unity offers no difficulty to his mind, though he is
conscious at first of the object itself rather than of any idea of number
associated with it. The concept of duality, also, is grasped with perfect
readiness. This concept is, in its simplest form, presented to the mind as
soon as the individual distinguishes himself from another person, though
the idea is still essentially concrete. Perhaps the first glimmering of any real
number thought in connection with 2 comes when the savage contrasts one
single object with another—or, in other words, when he first recognizes the
pair. At first the individuals composing the pair are simply “this one,” and
“that one,” or “this and that”; and his number system now halts for a time at
the stage - 75 -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 - 76 -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 - 77 -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 - 78 -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 -
79 -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. - 80 -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
- 81 -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
- 82 -
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 -
83 -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 - 84 -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 - 85 - × 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 - 86 -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 - 87 -with kan, tree; the sixth for counting canoes; and the
seventh for measures. The last seem to be composed with anon, hand.”139
The first ten numerals of each of these classes is given in the following
table:

No. Counting Flat Round Men Long Canoes Measures
Objects Objects Objects
1 gyak gak g'erel k'al k'awutskan k'amaet k'al
2 t'epqat t'epqat goupel t'epqadal gaopskan g'alpēeltk gulbel
3 guant guant gutle gulal galtskan galtskantk guleont
4 tqalpq tqalpq tqalpq tqalpqdal tqaapskan tqalpqsk tqalpqalont
5 kctōnc kctōnc kctōnc kcenecal k'etoentskan kctōonsk kctonsilont
6 k'alt k'alt k'alt k'aldal k'aoltskan k'altk k'aldelont
7 t'epqalt t'epqalt t'epqalt t'epqaldal t'epqaltskan t'epqaltk t'epqaldelont
8 guandalt yuktalt yuktalt yuktleadal ek'tlaedskan yuktaltk yuktaldelont
9 kctemac kctemac kctemac kctemacal kctemaestkan kctemack kctemasilont
10 gy'ap gy'ap kpēel kpal kpēetskan gy'apsk kpeont

Remarkable as this list may appear, it is by no means as extensive as
- 88 -
that derived from many of the other British Columbian tribes. The numerals
of the Shushwap, Stlatlumh, Okanaken, and other languages of this region
exist in several different forms, and can also be modified by any of the
innumerable suffixes of these tongues.140 To illustrate the almost illimitable
number of sets that may be formed, a table is given of “a few classes, taken
from the Heiltsuk dialect.141 It appears from these examples that the
number of classes is unlimited.”

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

One. Two. Three.
Together on beach. —— maalis ——
Together in house, etc. —— maalitl ——

Variation in numeral forms such as is exhibited in the above tables is
- 89 -
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,
- 90 -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 - 91 -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

- 92 -1. de = a going, i.e. a beginning. (Cf. the Zuñi töpinte, taken to start with.)
3. eto = the father (from the middle, or longest finger).

6. ade = the other going.
9. asieke = parting with the hands.
10. ewo = done.

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 - 93 -
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 - 94 -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 -
95 -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 - 96 -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, - 97 -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 - 98 -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 - 99 -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

- 100 - CHAPTER V.
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 - 101 -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 - 102 -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, - 103 -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 - 104 -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, - 105 -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 - 106 -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.

- 107 -Lake Kopperamana.
1. ngerna.
2. mondroo.
3. barkooloo.
4. mondroo mondroo = 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.

- 108 -Frazer's Island.
175
1. kalim.
2. bulla.
3. goorbunda.
4. bulla-bulla = 2-2.

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.

- 109 -Murray River, N.W. Bend.
183
1. mata.
2. rankool.
3. rankool mata = 2-1.
4. rankool rankool = 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.

- 110 -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:

- 111 -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, - 112 -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 India196 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 - 113 -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 - 114 -
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 - 115 -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 - 116 -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, is203

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
- 117 -
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, - 118 -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.
- 119 -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 Mocobi207 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 - 120 -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 - 121 -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

- 122 -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 - 123 -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 - 124 -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 saltee 1d. uno soldo.
dooe saltee 2d. due soldi.
tray saltee 3d. tre soldi.
quarterer saltee 4d. quattro soldi.
chinker saltee 5d. cinque soldi.
- 125 -say saltee 6d. sei soldi.
say oney saltee, or setter saltee 7d. sette soldi.
say dooe saltee, or otter saltee 8d. otto soldi.
say tray saltee, or nobba saltee 9d. nove soldi.
say quarterer saltee, or dacha saltee 10d. dieci soldi.
say chinker saltee or dacha oney saltee 11d. undici soldi.
oney beong 1s.
a beong say saltee 1s. 6d.
dooe beong say saltee, or madza caroon 2s. 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:

- 126 -5 ft. = five feet.
6 ft. = six feet.
9 ft. = nine feet.
10-1/2 ft. = a quarter less twain; i.e. a quarter of a fathom less than 2.
12 ft. = mark twain.
13-1/2 ft. = a quarter twain.
16-1/2 ft. = a quarter less three.
18 ft. = mark three.
19-1/2 ft. = a quarter three.
24 ft. = 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 - 127 -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 - 128 -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.
- 129 -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 - 130 -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. Kamassin.
8. siti-data = 2 × 4. 8. sin-the'de = 2 × 4.
9. nameaitjuma = another. 9. amithun = another.

But it would be childish to argue, from this fact alone, that either 4 or
- 131 -
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 - 132 -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 - 133 -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.

- 134 - CHAPTER VI.
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 - 135 -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.

- 136 -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 - 137 -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 - 138 -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.

Pikumbul
5. mulanbu.
10. bularin murra = belonging to the two hands.
- 139 -15. mulanba dinna = 5 toes added on (to the 10 fingers).
20. bularin dinna = belonging to the 2 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 - 140 -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
- 141 -
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.

- 142 -Canaque
242 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 - 143 -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.
- 144 -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 - 145 -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.

- 146 -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.

- 147 -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.

- 148 -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.

- 149 -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.

- 150 -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.

- 151 -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.

- 152 -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.

- 153 -Uea, Loyalty Island.
280
1. tahi.
2. lua.
3. tolu.
4. fa.
5. lima.
6. tahi.
7. lua.
8. tolu.
9. fa.
10. lima.

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.

- 154 -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.

- 155 -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.

- 156 -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.

- 157 -Kottisch
291
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.

- 158 -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 - 159 -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.

- 160 -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:

- 161 -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

- 162 -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.

- 163 -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.

- 164 -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.

- 165 -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.

- 166 -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
- 167 -
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.

- 168 -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 - 169 -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
- 170 -
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 - 171 -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.

- 172 -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 - 173 -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 - 174 -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
- 175 -
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.

- 176 - CHAPTER VII.
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 - 177 -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 - 178 -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,” - 179 -“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 - 180 -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.

- 181 -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
- 182 -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

- 183 -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 - 184 -
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).

- 185 -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.

- 186 -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
- 187 -
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 - 188 -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).

- 189 -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.

- 190 -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 - 191 -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.
- 192 -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 - 193 -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 -
194 -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 Coast Tribes
10. teya. 10. sham.
20. heng-inai. 20. heang-inai.
30. heng-inai-tain = 20 + 5 (couples). 30. heang-inai-tanai = 20 + 5 (couples).
40. au-inai = 2 × 20. 40. an-inai = 2 × 20.
100. tain-inai = 5 × 20. 100. tanai-inai = 5 × 20.
200. teya-inai = 10 × 20. 200. sham-inai = 10 × 20.
300. teya-tain-inai = (10 + 5) × 20. 300. heang-tanai-inai = (10 + 5) 20.
400. heng-teo. 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. - 195 -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 - 196 -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 - 197 -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 - 198 -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 - 199 -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.
- 200 -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 is364

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 - 201 -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.

- 202 -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 - 203 -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 - 204 -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. - 205 -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.

- 206 -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 - 207 -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 - 208 -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 - 209 -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 - 210 -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.

- 211 - INDEX OF AUTHORS.

Adam, L., 44, 159, 166, 175.
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.
Bellamy, E. W., 9.
Boas, F., 30, 45, 46, 65, 87, 88, 136, 163, 164, 171, 197, 198.
Bonwick, J., 24, 27, 107, 108.
Brinton, D. G., 2, 22, 46, 52, 57, 61, 111, 112, 140, 199, 200.
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.
Curr, E. M., 24–27, 104, 107–110, 112.
Cushing, F. H., 13, 48.

De Flacourt, 8, 9.
De Quincey, T., 35.
Deschamps, M., 28.
Dobrizhoffer, M., 71.

Dorsey, J. O., 59.
Du Chaillu, P. B., 66, 67, 150, 151.
Du Graty, A. M., 138.

Ellis, A. A., 64, 91.
Ellis, R., 37, 142.
Ellis, W., 83, 119.
Erskine, J. E., 153, 154.

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.
- 212 -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.

- 213 - INDEX OF SUBJECTS.

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.
- 214 -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.
- 215 -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.
- 216 -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.
- 217 -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.
- 218 -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.

- 219 - FOOTNOTES:

1 Brinton, D. G., Essays of an Americanist, p. 406; and American Race, p. 359.
2 This information I received from Dr. Brinton by letter.
3 Tylor, Primitive Culture, Vol. I. p. 240.
4 Nature, Vol. XXXIII. p. 45.
5 Spix and Martius, Travels in Brazil, Tr. from German by H. E. Lloyd, Vol. II. p.
255.
6 De Flacourt, Histoire de le grande Isle de Madagascar, ch. xxviii. Quoted by
Peacock, Encyc. Met., Vol. I. p. 393.
7 Bellamy, Elizabeth W., Atlantic Monthly, March, 1893, p. 317.
8 Grundriss der Sprachwissenschaft, Bd. III. Abt. i., p. 94.
9 Pruner-Bey, Bulletin de la Société d'Anthr. de Paris, 1861, p. 462.
10 “Manual Concepts,” Am. Anthropologist, 1892, p. 292.
11 Tylor, Primitive Culture, Vol. I. p. 245.
12 Op. cit., loc. cit.
13 “Aboriginal Inhabitants of Andaman Islands,” Journ. Anth. Inst., 1882, p. 100.
14 Morice, A., Revue d'Anthropologie, 1878, p. 634.
15 Macdonald, J., “Manners, Customs, etc., of South African Tribes,” Journ. Anthr.
Inst., 1889, p. 290. About a dozen tribes are enumerated by Mr. Macdonald: Pondos,
Tembucs, Bacas, Tolas, etc.
16 Codrington, R. H., Melanesians, their Anthropology and Folk-Lore, p. 353.
17 E.g. the Zuñis. See Cushing's paper quoted above.
18 Haddon, A. C., “Ethnography Western Tribes Torres Strait,” Journ. Anth. Inst.,
1889, p. 305. For a similar method, see Life in the Southern Isles, by W. W. Gill.
19 Tylor, Primitive Culture, Vol. I. p. 246.

20 Brinton, D. G., Letter of Sept. 23, 1893.
21 Ibid. The reference for the Mbocobi, infra, is the same. See also Brinton's
American Race, p. 361.
22 Tylor, Primitive Culture, Vol. I. p. 243.
23 Op. cit., loc. cit.
24 Hyades, Bulletin de la Société d'Anthr. de Paris, 1887, p. 340.
25 Wiener, C., Pérou et Bolivie, p. 360.
26 Marcoy, P., Travels in South America, Vol. II p. 47. According to the same
authority, most of the tribes of the Upper Amazon cannot count above 2 or 3 except by
reduplication.
27 Op. cit., Vol. II. p. 281.
28 Glossaria Linguarum Brasiliensium. Bororos, p. 15; Guachi, p. 133; Carajas, p.
265.
29 Curr, E. M., The Australian Race, Vol. I. p. 282. The next eight lists are, in
order, from I. p. 294, III. p. 424, III. p. 114, III. p. 124, II. p. 344, II. p. 308, I. p. 314,
III. p. 314, respectively.
30 Bonwick, J., The Daily Life and Origin of the Tasmanians, p. 144.
31 Latham, Comparative Philology, p. 336.
32 The Australian Race, Vol. I. p. 205.
33 Mackenzie, A., “Native Australian Langs.,” Journ. Anthr. Inst., 1874, p. 263.
34 Curr, The Australian Race, Vol. II. p. 134. The next four lists are from II. p. 4, I.
p. 322, I. p. 346, and I. p. 398, respectively.
35 Curr, op. cit., Vol. III. p. 50.
36 Op. cit., Vol. III. p. 236.
37 Müller, Sprachwissenschaft. II. i. p. 23.
38 Op. cit., II. i. p. 31.
39 Bonwick, op. cit., p. 143.
40 Curr, op. cit., Vol. I. p. 31.
41 Deschamps, L'Anthropologie, 1891, p. 318.
42 Man, E. H. Aboriginal Inhabitants of the Andaman Islands, p. 32.
43 Müller, Sprachwissenschaft, I. ii. p. 29.
44 Oldfield, A., Tr. Eth. Soc. Vol. III. p. 291.
45 Bancroft, H. H., Native Races, Vol. I. p. 564.

46 “Notes on Counting, etc., among the Eskimos of Point Barrow.” Am. Anthrop.,
1890, p. 38.
47 Second Voyage, p. 556.
48 Personal Narrative, Vol. I. p. 311.
49 Burton, B. F., Mem. Anthr. Soc. of London, Vol. I. p. 314.
50 Confessions. In collected works, Edinburgh, 1890, Vol. III. p. 337.
51 Ellis, Robert, On Numerals as Signs of Primeval Unity. See also Peruvia
Scythia, by the same author.
52 Stanley, H. M., In Darkest Africa, Vol. II. p. 493.
53 Stanley, H. M., Through the Dark Continent, Vol. II. p. 486.
54 Haumontè, Parisot, Adam, Grammaire et Vocabulaire de la Langue Taensa, p.
20.
55 Chamberlain, A. F., Lang. of the Mississaga Indians of Skugog. Vocab.
56 Boas, Fr., Sixth Report on the Indians of the Northwest, p. 105.
57 Beauregard, O., Bulletin de la Soc. d'Anthr. de Paris, 1886, p. 526.
58 Ray, S. H., Journ. Anthr. Inst., 1891, p. 8.
59 Op. cit., p. 12.
60 Müller, Sprachwissenschaft, IV. i. p. 136.
61 Brinton, The Maya Chronicles, p. 50.
62 Trumbull, On Numerals in Am. Ind. Lang., p. 35.
63 Boas, Fr. This information was received directly from Dr. Boas. It has never
before been published.
64 Bancroft, H. H., Native Races, Vol. II. p. 753. See also p. 199, infra.
65 Mann, A., “Notes on the Numeral Syst. of the Yoruba Nation,” Journ. Anth.
Inst., 1886, p. 59, et seq.
66 Müller, Sprachwissenschaft, IV. i. p. 202.
67 Trumbull, J. H., On Numerals in Am. Ind. Langs., p. 11.
68 Cushing, F. H., “Manual Concepts,” Am. Anthr., 1892, p. 289.
69 Grimm, Geschichte der deutschen Sprache, Vol. I. p. 239.
70 Murdoch, J., American Anthropologist, 1890, p. 39.
71 Kleinschmidt, S., Grammatik der Grönlandischen Sprache, p. 37.
72 Brinton, The Arawak Lang. of Guiana, p. 4.

73 Petitot, E., Dictionnaire de la langue Dènè-Dindjie, p. lv.
74 Gilij, F. S., Saggio di Storia Am., Vol. II. p. 333.
75 Müller, Sprachwissenschaft, II. i. p. 389.
76 Op. cit., p. 395.
77 Müller, Sprachwissenschaft, II. i. p. 438.
78 Peacock, “Arithmetic,” in Encyc. Metropolitana, 1, p. 480.
79 Brinton, D. G., “The Betoya Dialects,” Proc. Am. Philos. Soc., 1892, p. 273.
80 Ridley, W., “Report on Australian Languages and Traditions.” Journ. Anth. Inst.,
1873, p. 262.
81 Gatschet, “Gram. Klamath Lang.” U. S. Geog. and Geol. Survey, Vol. II. part 1,
pp. 524 and 536.
82 Letter of Nov. 17, 1893.
83 Müller, Sprachwissenschaft, II. i. p. 439.
84 Hale, “Indians of No. West. Am.,” Tr. Am. Eth. Soc., Vol. II. p. 82.
85 Brinton, D. G., Studies in So. Am. Native Languages, p. 25.
86 Tr. Am. Philological Association, 1874, p. 41.
87 Tylor, Primitive Culture, Vol. I. p. 251.
88 Müller, Sprachwissenschaft, IV. i. p. 27.
89 See infra, Chapter VII.
90 Ellis, A. B., Ewe Speaking Peoples, etc., p. 253.
91 Tylor, Primitive Culture, Vol. I. p. 256.
92 Stanley, In Darkest Africa, Vol. II. p. 493.
93 Chamberlain, A. F., Proc. Brit. Ass. Adv. of Sci., 1892, p. 599.
94 Boas, Fr., “Sixth Report on Northwestern Tribes of Canada,” Proc. Brit. Ass.
Adv. Sci., 1890, p. 657.
95 Hale, H., “Indians of Northwestern Am.,” Tr. Am. Eth. Soc., Vol. II. p. 88.
96 Op. cit., p. 95.
97 Müller, Sprachwissenschaft, II. ii. p. 147.
98 Schoolcraft, Archives of Aboriginal Knowledge, Vol. IV. p. 429.
99 Du Chaillu, P. B., Tr. Eth. Soc., London, Vol. I. p. 315.
100 Latham, R. G., Essays, chiefly Philological and Ethnographical, p. 247. The
above are so unlike anything else in the world, that they are not to be accepted without

careful verification.
101 Pott, Zählmethode, p. 45.
102 Gatschet, A. S., The Karankawa Indians, the Coast People of Texas. The
meanings of 6, 7, 8, and 9 are conjectural with me.
103 Stanley, H. M., In Darkest Africa, Vol. II. p. 492.
104 Müller, Sprachwissenschaft, II. i. p. 317.
105 Toy, C. H., Trans. Am. Phil. Assn., 1878, p. 29.
106 Burton, R. F., Mem. Anthrop. Soc. of London. 1, p. 314. In the illustration
which follows, Burton gives 6820, instead of 4820; which is obviously a misprint.
107 Dobrizhoffer, History of the Abipones, Vol. II. p. 169.
108 Sayce, A. H., Comparative Philology, p. 254.
109 Tr. Eth. Society of London , Vol. III. p. 291.
110 Ray, S. H., Journ. Anthr. Inst., 1889, p. 501.
111 Stanley, In Darkest Africa, Vol. II. p. 492.
112 Op. cit., loc. cit.
113 Tylor, Primitive Culture, Vol. I. p. 249.
114 Müller, Sprachwissenschaft, IV. i. p. 36.
115 Martius, Glos. Ling. Brasil., p. 271.
116 Tylor, Primitive Culture, Vol. I. p. 248.
117 Roth, H. Ling, Aborigines of Tasmania, p. 146.
118 Lull, E. P., Tr. Am. Phil, Soc., 1873, p. 108.
119 Ray, S. H. “Sketch of Api Gram.,” Journ. Anthr. Inst., 1888, p. 300.
120 Kleinschmidt, S., Grammatik der Grönlandischen Spr., p. 39.
121 Müller, Sprachwissenschaft, I. ii. p. 184.
122 Op. cit., I. ii. p. 18, and II. i. p. 222.
123 Squier, G. E., Nicaragua, Vol. II. p. 326.
124 Schoolcraft, H. R., Archives of Aboriginal Knowledge, Vol. II. p. 208.
125 Tylor, Primitive Culture, Vol. I. p. 264.
126 Goedel, “Ethnol. des Soussous,” Bull. de la Soc. d'Anthr. de Paris, 1892, p.
185.
127 Ellis, W., History of Madagascar, Vol. I. p. 507.

128 Beauregard, O., Bull. de la Soc. d'Anthr. de Paris, 1886, p. 236.
129 Schoolcraft, H. R., Archives of Aboriginal Knowledge, Vol. II. p. 207.
130 Tylor, Primitive Culture, Vol. I. p. 249.
131 Op. cit. Vol. I. p. 250.
132 Peacock, Encyc. Metropolitana, 1, p. 478.
133 Op. cit., loc. cit.
134 Schoolcraft, H. R., Archives of Aboriginal Knowledge, Vol. II. p. 213.
135 Op. cit., p. 216.
136 Op. cit., p. 206.
137 Mariner, Gram. Tonga Lang., last part of book. [Not paged.]
138 Morice, A. G., “The Déné Langs,” Trans. Can. Inst., March 1890, p. 186.
139 Boas, Fr., “Fifth Report on the Northwestern Tribes of Canada,” Proc. Brit.
Ass. Adv. of Science, 1889, p. 881.
140 Do. Sixth Rep., 1890, pp. 684, 686, 687.
141 Op. cit., p. 658.
142 Bancroft, H. H., Native Races, Vol. II. p. 499.
143 Tr. Ethnological Soc. of London, Vol. IV. p. 92.
144 Any Hebrew lexicon.
145 Schröder, P., Die Phönizische Sprache, p. 184 et seq.
146 Müller, Sprachwissenschaft, II. ii. p. 147.
147 On Numerals in Am. Indian Languages.
148 Ellis, A. B., Ewe Speaking Peoples, etc., p. 253. The meanings here given are
partly conjectural.
149 Pott, Zählmethode, p. 29.
150 Schoolcraft, op. cit., Vol. IV. p. 429.
151 Trumbull, op. cit.
152 Chamberlain, A. F., Lang, of the Mississaga Indians, Vocab.
153 Crawfurd, Hist. Ind. Archipelago, 1, p. 258.
154 Hale, H., Eth. and Philol., Vol. VII.; Wilkes, Expl. Expedition, Phil. 1846, p.
172.
155 Crawfurd, op. cit., 1, p. 258.

156 Op. cit., loc. cit.
157 Bancroft, H. H., Native Races, Vol. II. p. 498.
158 Vignoli, T., Myth and Science, p. 203.
159 Codrington, R. H., The Melanesian Languages, p. 249.
160 Op. cit., loc. cit.
161 Codrington, R. H., The Melanesian Languages, p. 249.
162 Wickersham, J., “Japanese Art on Puget Sound,” Am. Antiq., 1894, p. 79.
163 Codrington, R. H., op. cit., p. 250.
164 Tylor, Primitive Culture, Vol. I. p. 252.
165 Compare a similar table by Chase, Proc. Amer. Philos. Soc., 1865, p. 23.
166 Leibnitzii Opera, III. p. 346.
167 Pruner-Bey, Bulletin de la Soc. d'Anthr. de Paris, 1860, p. 486.
168 Curr, E. M., The Australian Race, Vol. I. p. 32.
169 Haddon, A. C., “Western Tribes of the Torres Straits,” Journ. Anthr. Inst.,
1889, p. 303.
170 Taplin, Rev. G., “Notes on a Table of Australian Languages,” Journ. Anthr.
Inst., 1872, p. 88. The first nine scales are taken from this source.
171 Latham, R. G., Comparative Philology, p. 352.
172 It will be observed that this list differs slightly from that given in Chapter II.
173 Curr, E. M., The Australian Race, Vol. III. p. 684.
174 Bonwick, Tasmania, p. 143.
175 Lang, J. D., Queensland, p. 435.
176 Bonwick, Tasmania, p. 143.
177 Müller, Sprachwissenschaft, II. i. p. 58.
178 Op. cit., II. i. p. 70.
179 Op. cit., II. i. p. 23.
180 Barlow, H., “Aboriginal Dialects of Queensland,” Journ. Anth. Inst., 1873, p.
171.
181 Curr, E. M., The Australian Race, Vol. II. p. 26.
182 Op. cit., Vol. II. p. 208.
183 Op. cit., Vol. II. p. 278.

184 Op. cit., Vol. II. p. 288.
185 Op. cit., Vol. I. p. 258.
186 Op. cit., Vol. I. p. 316.
187 Op. cit., Vol. III. p. 32. The next ten lists are taken from the same volume, pp.
282, 288, 340, 376, 432, 506, 530, 558, 560, 588, respectively.
188 Brinton, The American Race, p. 351.
189 Martius, Glossaria Ling. Brazil., p. 307.
190 Op. cit., p. 148.
191 Müller, Sprachwissenschaft, II. i. p. 438.
192 Peacock, “Arithmetic,” Encyc. Metropolitana, 1, p. 480.
193 Brinton, Studies in So. Am. Native Langs., p. 67.
194 Op. cit., loc. cit.
195 Brinton, Studies in So. Am. Native Langs., p. 67. The meanings of the numerals
are from Peacock, Encyc. Metropolitana, 1, p. 480.
196 Mason, Journ. As. Soc. of Bengal, Vol. XXVI. p. 146.
197 Curr, E. M., The Australian Race, Vol. III. p. 108.
198 Bancroft, H. H., Native Races, Vol. I. p. 274.
199 Clarke, Hyde, Journ. Anthr. Inst., 1872, p. clvii. In the article from which this
is quoted, no evidence is given to substantiate the assertion made. It is to be received
with great caution.
200 Hale, H., Wilkes Exploring Expedition, Vol. VII. p. 172.
201 Op. cit., p. 248.
202 Hale, Ethnography and Philology, p. 247.
203 Loc. cit.
204 Ellis, Polynesian Researches, Vol. IV. p. 341.
205 Gill, W. W., Myths and Songs of the South Pacific, p. 325.
206 Peacock, “Arithmetic,” Encyc. Metropolitana, 1, p. 479.
207 Peacock, Encyc. Metropolitana, 1, p. 480.
208 Sprachverschiedenheit, p. 30.
209 Crawfurd, History of the Indian Archipelago, Vol. I. p. 256.
210 Pott, Zählmethode, p. 39.
211 Op. cit., p. 41.

212 Müller, Sprachwissenschaft, II. i. p. 317. See also Chap. III., supra.
213 Long, S. H., Expedition, Vol. II. p. lxxviii.
214 Martius, Glossaria Ling. Brasil., p. 246.
215 Hale, Ethnography and Philology, p. 434.
216 Müller, Sprachwissenschaft, II. ii. p. 82.
217 The information upon which the above statements are based was obtained from
Mr. W. L. Williams, of Gisborne, N.Z.
218 Primitive Culture, Vol. I. p. 268.
219 Ralph, Julian, Harper's Monthly, Vol. 86, p. 184.
220 Lappenberg, J. M., History of Eng. under the Anglo-Saxon Kings, Vol. I. p. 82.
221 The compilation of this table was suggested by a comparison found in the
Bulletin Soc. Anth. de Paris, 1886, p. 90.
222 Hale, Ethnography and Philology, p. 126.
223 Müller, Sprachwissenschaft, II. ii. p. 183.
224 Bachofen, J. J., Antiquarische Briefe, Vol. I. pp. 101–115, and Vol. II. pp. 1–
90.
225 An extended table of this kind may be found in the last part of Nystrom's
Mechanics.
226 Schubert, H., quoting Robert Flegel, in Neumayer's Anleitung zu
Wissenschaftlichen Beobachtung auf Reisen, Vol. II. p. 290.
227 These numerals, and those in all the sets immediately following, except those
for which the authority is given, are to be found in Chapter III.
228 Codrington, The Melanesian Languages, p. 222.
229 Müller, Sprachwissenschaft, II. ii. p. 83.
230 Op. cit., I. ii. p. 55. The next two are the same, p. 83 and p. 210. The meaning
given for the Bari puök is wholly conjectural.
231 Gallatin, “Semi-civilized Nations,” Tr. Am. Eth. Soc., Vol. I. p. 114.
232 Müller, Sprachwissenschaft, II. ii. p. 80. Erromango, the same.
233 Boas, Fr., Proc. Brit. Ass'n. Adv. Science, 1889, p. 857.
234 Hankel, H., Geschichte der Mathematik, p. 20.
235 Murdoch, J., “Eskimos of Point Barrow,” Am. Anthr., 1890, p. 40.
236 Martius, Glos. Ling. Brasil., p. 360.
237 Du Graty, A. M., La République du Paraguay, p. 217.

238 Codrington, The Melanesian Languages, p. 221.
239 Müller, Sprachwissenschaft, II. i. p. 363.
240 Spurrell, W., Welsh Grammar, p. 59.
241 Olmos, André de, Grammaire Nahuatl ou Mexicaine, p. 191.
242 Moncelon, Bull. Soc. d'Anthr. de Paris, 1885, p. 354. This is a purely digital
scale, but unfortunately M. Moncelon does not give the meanings of any of the
numerals except the last.
243 Ellis, Peruvia Scythia, p. 37. Part of these numerals are from Martius, Glos.
Brasil., p. 210.
244 Codrington, The Melanesian Languages, p. 236.
245 Schweinfurth, G., Linguistische Ergebnisse einer Reise nach Centralafrika, p.
25.
246 Park, M., Travels in the Interior Districts of Africa, p. 8.
247 Pott, Zählmethode, p. 37.
248 Op. cit., p. 39.
249 Müller, Sprachwissenschaft, IV. i. p. 101. The Kru scale, kindred with the
Basa, is from the same page.
250 Park, in Pinkerton's Voyages and Travels, Vol. XVI. p. 902.
251 Park, Travels, Vol. I. p. 16.
252 Schweinfurth, G., Linguistische Ergebnisse einer Reise nach Centralafrika, p.
78.
253 Park, Travels, Vol. I. p. 58.
254 Goedel, “Ethnol. des Soussous,” Bull. Soc. Anth. Paris, 1892, p. 185.
255 Müller, Sprachwissenschaft, I. ii. p. 114. The Temne scale is from the same
page. These two languages are closely related.
256 Op. cit., I. ii. p. 155.
257 Op. cit., I. ii. p. 55.
258 Long, C. C., Central Africa, p. 330.
259 Müller, Sprachwissenschaft, IV. i. p. 105.
260 Pott, Zählmethode, p. 41.
261 Müller, op. cit., I. ii. p. 140.
262 Müller, Sprachwissenschaft, IV. i. p. 81.
263 Pott, Zählmethode, p. 41.

264 Müller, op. cit., I. ii., p. 210.
265 Pott, Zählmethode, p. 42.
266 Schweinfurth, Linguistische Ergebnisse, p. 59.
267 Müller, Sprachwissenschaft, I. ii. p. 261. The “ten” is not given.
268 Stanley, Through the Dark Continent, Vol. II. p. 490. Ki-Nyassa, the same
page.
269 Müller, op. cit., I. ii. p. 261.
270 Du Chaillu, Adventures in Equatorial Africa, p. 534.
271 Müller, Sprachwissenschaft, III. i. p. 65.
272 Du Chaillu, Adventures in Equatorial Africa, p. 533.
273 Müller, op. cit., III. ii. p. 77.
274 Balbi, A., L'Atlas Eth., Vol. I. p. 226. In Balbi's text 7 and 8 are ansposed. Taru
for 5 is probably a misprint for tana.
275 Du Chaillu, op. cit., p. 533. The next scale is op. cit., p. 534.
276 Beauregard, O., Bull. Soc. Anth. de Paris, 1886, p. 526.
277 Pott, Zählmethode, p. 46.
278 Op. cit., p. 48.
279 Turner, Nineteen Years in Polynesia, p. 536.
280 Erskine, J. E., Islands of the Western Pacific, p. 341.
281 Op. cit., p. 400.
282 Codrington, Melanesian Languages, pp. 235, 236.
283 Peacock, Encyc. Met., Vol. 1. p. 385. Peacock does not specify the dialect.
284 Erskine, Islands of the Western Pacific, p. 360.
285 Turner, G., Samoa a Hundred Years Ago, p. 373. The next three scales are from
the same page of this work.
286 Codrington, Melanesian Languages, p. 235. The next four scales are from the
same page. Perhaps the meanings of the words for 6 to 9 are more properly “more 1,”
“more 2,” etc. Codrington merely indicates their significations in a general way.
287 Hale, Ethnography and Philology, p. 429. The meanings of 6 to 9 in this and
the preceding are my conjectures.
288 Müller, Sprachwissenschaft, IV. i. p. 124.
289 Aymonier, E., Dictionnaire Francaise-Cambodgien.

290 Müller, Op. cit., II. i. p. 139.
291 Müller, Sprachwissenschaft, II. i. p. 123.
292 Wells, E. R., Jr., and John W. Kelly, Bureau of Ed., Circ. of Inf., No. 2, 1890.
293 Pott, Zählmethode, p. 57.
294 Müller, Op. cit., II. i. p. 161.
295 Petitot, Vocabulaire Française Esquimau, p. lv.
296 Müller, Sprachwissenschaft, II. i. p. 253.
297 Müller, Op. cit., II. I.i. p. 179, and Kleinschmidt, Grönlandisches Grammatik.
298 Adam, L., Congres Int. des Am., 1877, p. 244 (see p. 162 infra).
299 Gallatin, “Synopsis of Indian Tribes,” Trans. Am. Antq. Soc., 1836, p. 358. The
next fourteen lists are, with the exception of the Micmac, from the same collection.
The meanings are largely from Trumbull, op. cit.
300 Schoolcraft, Archives of Aboriginal Knowledge, Vol. II. p. 211.
301 Schoolcraft, Archives of Aboriginal Knowledge, Vol. V. p. 587.
302 In the Dakota dialects 10 is expressed, as here, by a word signifying that the
fingers, which have been bent down in counting, are now straightened out.
303 Boas, Fifth Report B. A. A. S., 1889. Reprint, p. 61.
304 Boas, Sixth Report B. A. A. S., 1890. Reprint, p. 117. Dr. Boas does not give
the meanings assigned to 7 and 8, but merely states that they are derived from 2 and 3.
305 Op. cit., p. 117. The derivations for 6 and 7 are obvious, but the meanings are
conjectural.
306 Boas, Sixth Report B. A. A. S., 1889. Reprint, pp. 158, 160. The meanings
assigned to the Tsimshian 8 and to Bilqula 6 to 8 are conjectural.
307 Hale, Ethnography and Philology, p. 619.
308 Op. cit., loc. cit.
309 Hale, Ethnography and Philology, p. 619.
310 Müller, Sprachwissenschaft, II. i. p. 436.
311 Op. cit., IV. i. p. 167.
312 Op. cit., II. i. p. 282.
313 Op. cit., II. i. p. 287. The meanings given for the words for 7, 8, 9 are
conjectures of my own.
314 Müller, Sprachwissenschaft, II. i. p. 297.
315 Pott, Zählmethode, p. 90.

316 Müller, op. cit., II. i. p. 379.
317 Gallatin, “Semi-Civilized Nations of Mexico and Central America,” Tr. Am.
Ethn. Soc., Vol. I. p. 114.
318 Adam, Lucien, Congres Internationale des Americanistes, 1877, Vol. II. p.
244.
319 Müller, Sprachwissenschaft, II. i. p. 395. I can only guess at the meanings of 6
to 9. They are obviously circumlocutions for 5-1, 5-2, etc.
320 Op. cit., p. 438. Müller has transposed these two scales. See Brinton's Am.
Race, p. 358.
321 Marcoy, P., Tour du Monde, 1866, 2ème sem. p. 148.
322 Op. cit., p. 132. The meanings are my own conjectures.
323 An elaborate argument in support of this theory is to be found in Hervas'
celebrated work, Arithmetica di quasi tutte le nazioni conosciute.
324 See especially the lists of Hale, Gallatin, Trumbull, and Boas, to which
references have been given above.
325 Thiel, B. A., “Vocab. der Indianier in Costa Rica,” Archiv für Anth., xvi. p.
620.
326 These three examples are from A. R. Wallace's Narrative of Travels on the
Amazon and Rio Negro, vocab. Similar illustrations may be found in Martius' Glos.
Brasil.
327 Martius, Glos. Brasil., p. 176.
328 Adam, L., Congres International des Americanistes, 1877, Vol. II. p. 244.
Given also supra, p. 53.
329 O'Donovan, Irish Grammar, p. 123.
330 Armstrong, R. A., Gaelic Dict., p. xxi.
331 Spurrell, Welsh Dictionary.
332 Kelly, Triglot Dict., pub. by the Manx Society.
333 Guillome, J., Grammaire Française-Bretonne, p. 27.
334 Gröber, G., Grundriss der Romanischen Philologie, Bd. I. p. 309.
335 Pott, Zählmethode, p. 88.
336 Van Eys, Basque Grammar, p. 27.
337 Pott, Zählmethode, p. 101.
338 Op. cit., p. 78.
339 Müller, Sprachwissenschaft, I. ii. p. 124.

340 Op. cit., p. 155.
341 Op. cit., p. 140.
342 Op. cit., loc. cit.
343 Schweinfurth, Reise nach Centralafrika, p. 25.
344 Müller, Sprachwissenschaft, IV. i. p. 83.
345 Op. cit., IV. i. p. 81.
346 Op. cit., I. ii. p. 166.
347 Long, C. C., Central Africa, p. 330.
348 Peacock, Encyc. Met., Vol. I. p. 388.
349 Müller, Sprachwissenschaft, III. ii. p. 64. The next seven scales are from op.
cit., pp. 80, 137, 155, 182, 213.
350 Pott, Zählmethode, p. 83.
351 Op. cit., p. 83,—Akari, p. 84; Circassia, p. 85.
352 Müller, Sprachwissenschaft, II. i. p. 140.
353 Pott, Zählmethode, p. 87.
354 Müller, Sprachwissenschaft, II. ii. p. 346.
355 Op. cit., III. i. p. 130.
356 Man, E. H., “Brief Account of the Nicobar Islands,” Journ. Anthr. Inst., 1885,
p. 435.
357 Wells, E. R., Jr., and Kelly, J. W., “Eng. Esk. and Esk. Eng. Vocab.,” Bureau of
Education Circular of Information, No. 2, 1890, p. 65.
358 Petitot, E., Vocabulaire Française Esquimau, p. lv.
359 Boas, Fr., Proc. Brit. Ass. Adv. Sci., 1889, p. 857.
360 Boas, Sixth Report on the Northwestern Tribes of Canada, p. 117.
361 Boas, Fr., Fifth Report on the Northwestern Tribes of Canada, p. 85.
362 Gallatin, Semi-Civilized Nations, p. 114. References for the next two are the
same.
363 Bancroft, H. H., Native Races of the Pacific States, Vol. II. p. 763. The
meanings are from Brinton's Maya Chronicles, p. 38 et seq.
364 Brinton, Maya Chronicles, p. 44.
365 Siméon Rémi, Dictionnaire de la langue nahuatl, p. xxxii.

366 An error occurs on p. xxxiv of the work from which these numerals are taken,
which makes the number in question appear as 279,999,999 instead of 1,279,999,999.
367 Gallatin, “Semi-Civilized Nations of Mexico and Central America,” Tr. Am.
Ethn. Soc. Vol. I. p. 114.
368 Pott, Zählmethode, p. 89. The Totonacos were the first race Cortez encountered
after landing in Mexico.
369 Op. cit., p. 90. The Coras are of the Mexican state of Sonora.
370 Gallatin, Semi-Civilized Nations, p. 114.
371 Humboldt, Recherches, Vol. II. p. 112.
372 Squier, Nicaragua, Vol. II. p. 326.
373 Gallatin, Semi-Civilized Nations, p. 57.

*** END OF THE PROJECT GUTENBERG EBOOK THE NUMBER
CONCEPT: ITS ORIGIN AND DEVELOPMENT ***

Updated editions will replace the previous one—the old editions will be
renamed.

Creating the works from print editions not protected by U.S. copyright law
means that no one owns a United States copyright in these works, so the
Foundation (and you!) can copy and distribute it in the United States
without permission and without paying copyright royalties. Special rules,
set forth in the General Terms of Use part of this license, apply to copying
and distributing Project Gutenberg™ electronic works to protect the
PROJECT GUTENBERG™ concept and trademark. Project Gutenberg is a
registered trademark, and may not be used if you charge for an eBook,
except by following the terms of the trademark license, including paying
royalties for use of the Project Gutenberg trademark. If you do not charge
anything for copies of this eBook, complying with the trademark license is
very easy. You may use this eBook for nearly any purpose such as creation
of derivative works, reports, performances and research. Project Gutenberg
eBooks may be modified and printed and given away—you may do
practically ANYTHING in the United States with eBooks not protected by
U.S. copyright law. Redistribution is subject to the trademark license,
especially commercial redistribution.

START: FULL LICENSE

THE FULL PROJECT GUTENBERG™ LICENSE
PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK

To protect the Project Gutenberg™ mission of promoting the free
distribution of electronic works, by using or distributing this work (or any
other work associated in any way with the phrase “Project Gutenberg”), you
agree to comply with all the terms of the Full Project Gutenberg License
available with this file or online at www.gutenberg.org/license.

Section 1. General Terms of Use and Redistributing
Project Gutenberg electronic works

1.A. By reading or using any part of this Project Gutenberg electronic work,
you indicate that you have read, understand, agree to and accept all the
terms of this license and intellectual property (trademark/copyright)
agreement. If you do not agree to abide by all the terms of this agreement,
you must cease using and return or destroy all copies of Project Gutenberg
electronic works in your possession. If you paid a fee for obtaining a copy
of or access to a Project Gutenberg electronic work and you do not agree to
be bound by the terms of this agreement, you may obtain a refund from the
person or entity to whom you paid the fee as set forth in paragraph 1.E.8.

1.B. “Project Gutenberg” is a registered trademark. It may only be used on
or associated in any way with an electronic work by people who agree to be
bound by the terms of this agreement. There are a few things that you can
do with most Project Gutenberg electronic works even without complying
with the full terms of this agreement. See paragraph 1.C below. There are a
lot of things you can do with Project Gutenberg electronic works if you
follow the terms of this agreement and help preserve free future access to
Project Gutenberg electronic works. See paragraph 1.E below.

1.C. The Project Gutenberg Literary Archive Foundation (“the Foundation”
or PGLAF), owns a compilation copyright in the collection of Project
Gutenberg electronic works. Nearly all the individual works in the
collection are in the public domain in the United States. If an individual
work is unprotected by copyright law in the United States and you are

located in the United States, we do not claim a right to prevent you from
copying, distributing, performing, displaying or creating derivative works
based on the work as long as all references to Project Gutenberg are
removed. Of course, we hope that you will support the Project Gutenberg
mission of promoting free access to electronic works by freely sharing
Project Gutenberg works in compliance with the terms of this agreement for
keeping the Project Gutenberg name associated with the work. You can
easily comply with the terms of this agreement by keeping this work in the
same format with its attached full Project Gutenberg License when you
share it without charge with others.

1.D. The copyright laws of the place where you are located also govern
what you can do with this work. Copyright laws in most countries are in a
constant state of change. If you are outside the United States, check the
laws of your country in addition to the terms of this agreement before
downloading, copying, displaying, performing, distributing or creating
derivative works based on this work or any other Project Gutenberg work.
The Foundation makes no representations concerning the copyright status
of any work in any country other than the United States.

1.E. Unless you have removed all references to Project Gutenberg:

1.E.1. The following sentence, with active links to, or other immediate
access to, the full Project Gutenberg License must appear prominently
whenever any copy of a Project Gutenberg work (any work on which the
phrase “Project Gutenberg” appears, or with which the phrase “Project
Gutenberg” is associated) is accessed, displayed, performed, viewed, copied
or distributed:

This eBook is for the use of anyone anywhere in the United States
and most other parts of the world at no cost and with almost no
restrictions whatsoever. You may copy it, give it away or re-use it
under the terms of the Project Gutenberg™ License included with
this eBook or online at www.gutenberg.org. If you are not located in
the United States, you will have to check the laws of the country
where you are located before using this eBook.