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Title: The Number Concept
       Its Origin and Development

Author: Levi Leonard Conant

Release Date: August 5, 2005 [EBook #16449]

Language: English

Character set encoding: ISO-8859-1

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[*Transcriber's Note:
The following errors found in the original have been left as is.
Chapter I, 14th paragraph:
  drop double quote before 'It is said';
Chapter IV, 1st paragraph:
  'so similar than' read 'so similar that';
Chapter IV, table of Hebrew numerals (near footnote 144):
  insert comma after 'shemoneh';
Chapter V, table of Tahuatan numerals (near footnote 201):
  'tahi,' read 'tahi.';
Same table:
  ' 20,000.  tufa' read '200,000.  tufa';
Chapter VI, table of Bagrimma numerals (near footnote 259):
  'marta = 5 + 2' read 'marta = 5 + 3';
Same table:
  'do-so = [5] + 3' read 'do-so = [5] + 4';
Chapter VII, table of Nahuatl numerals (near footnote 365):
  '90-10' read '80-10';
In paragraph following that table:
  '+ (15 + 4) × 400 × 800' read
  '(15 + 4) × 20 × 400 × 8000 + (15 + 4) × 400 × 8000';
In text of footnote 297:
  'II. I. p. 179' read 'II. i. p. 179';
*]


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






THE NUMBER CONCEPT



ITS ORIGIN AND DEVELOPMENT



BY
LEVI LEONARD CONANT, PH.D.
ASSOCIATE PROFESSOR OF MATHEMATICS IN THE WORCESTER
POLYTECHNIC INSTITUTE



New York
MACMILLAN AND CO.
AND LONDON
1931



COPYRIGHT, 1896,
BY THE MACMILLAN COMPANY.






COPYRIGHT, 1924,
BY EMMA B. CONANT.






All rights reserved--no part of this book may be reproduced in any form
without permission in writing from the publisher.






Set up and electrotyped. Published July, 1896.



Norwood Press
J.S. Cushing Co.--Berwick & Smith Co.
Norwood, Mass., U.S.A.





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 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.





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 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 the numbers 1 and 2, or 1,
2, and 3. Examples of this poverty of number knowledge are found among the
forest tribes of Brazil, the native races of Australia and elsewhere, and
they are considered in some detail in the next chapter. At first thought it
seems quite inconceivable that any human being should be destitute of the
power of counting beyond 2. But such is the case; and in a few instances
languages have been found to be absolutely destitute of pure numeral words.
The Chiquitos of Bolivia had no real numerals whatever,[1] but expressed
their idea for "one" by the word _etama_, meaning alone. The Tacanas of the
same country have no numerals except those borrowed from Spanish, or from
Aymara or Peno, languages with which they have long been in contact.[2] A
few other South American languages are almost equally destitute of numeral
words. But even here, rudimentary as the number sense undoubtedly is, it is
not wholly lacking; and some indirect expression, or some form of
circumlocution, shows a conception of the difference between _one_ and
_two_, or at least, between _one_ and _many_.

These facts must of necessity deter the mathematician from seeking to push
his investigation too far back toward the very origin of number.
Philosophers have endeavoured to establish certain propositions concerning
this subject, but, as might have been expected, have failed to reach any
common ground of agreement. Whewell has maintained that "such propositions
as that two and three make five are necessary truths, containing in them an
element of certainty beyond that which mere experience can give." Mill, on
the other hand, argues that any such statement merely expresses a truth
derived from early and constant experience; and in this view he is heartily
supported by Tylor.[3] But why this question should provoke controversy, it
is difficult for the mathematician to understand. Either view would seem to
be correct, according to the standpoint from which the question is
approached. We know of no language in which the suggestion of number does
not appear, and we must admit that the words which give expression to the
number sense would be among the early words to be formed in any language.
They express ideas which are, at first, wholly concrete, which are of the
greatest possible simplicity, and which seem in many ways to be clearly
understood, even by the higher orders of the brute creation. The origin of
number would in itself, then, appear to lie beyond the proper limits of
inquiry; and the primitive conception of number to be fundamental with
human thought.

In connection with the assertion that the idea of number seems to be
understood by the higher orders of animals, the following brief quotation
from a paper by Sir John Lubbock may not be out of place: "Leroy ...
mentions a case in which a man was anxious to shoot a crow. 'To deceive
this suspicious bird, the plan was hit upon of sending two men to the watch
house, one of whom passed on, while the other remained; but the crow
counted and kept her distance. The next day three went, and again she
perceived that only two retired. In fine, it was found necessary to send
five or six men to the watch house to put her out in her calculation. The
crow, thinking that this number of men had passed by, lost no time in
returning.' From this he inferred that crows could count up to four.
Lichtenberg mentions a nightingale which was said to count up to three.
Every day he gave it three mealworms, one at a time. When it had finished
one it returned for another, but after the third it knew that the feast was
over.... There is an amusing and suggestive remark in Mr. Galton's
interesting _Narrative of an Explorer in Tropical South Africa_. After
describing the Demara's weakness in calculations, he says: 'Once while I
watched a Demara floundering hopelessly in a calculation on one side of me,
I observed, "Dinah," my spaniel, equally embarrassed on the other; she was
overlooking half a dozen of her new-born puppies, which had been removed
two or three times from her, and her anxiety was excessive, as she tried to
find out if they were all present, or if any were still missing. She kept
puzzling and running her eyes over them backwards and forwards, but could
not satisfy herself. She evidently had a vague notion of counting, but the
figure was too large for her brain. Taking the two as they stood, dog and
Demara, the comparison reflected no great honour on the man....' According
to my bird-nesting recollections, which I have refreshed by more recent
experience, if a nest contains four eggs, one may safely be taken; but if
two are removed, the bird generally deserts. Here, then, it would seem as
if we had some reason for supposing that there is sufficient intelligence
to distinguish three from four. An interesting consideration arises with
reference to the number of the victims allotted to each cell by the
solitary wasps. One species of Ammophila considers one large caterpillar of
_Noctua segetum_ enough; one species of Eumenes supplies its young with
five victims; another 10, 15, and even up to 24. The number appears to be
constant in each species. How does the insect know when her task is
fulfilled? Not by the cell being filled, for if some be removed, she does
not replace them. When she has brought her complement she considers her
task accomplished, whether the victims are still there or not. How, then,
does she know when she has made up the number 24? Perhaps it will be said
that each species feels some mysterious and innate tendency to provide a
certain number of victims. This would, under no circumstances, be any
explanation; but it is not in accordance with the facts. In the genus
Eumenes the males are much smaller than the females.... If the egg is male,
she supplies five; if female, 10 victims. Does she count? Certainly this
seems very like a commencement of arithmetic."[4]

Many writers do not agree with the conclusions which Lubbock reaches;
maintaining that there is, in all such instances, a perception of greater
or less quantity rather than any idea of number. But a careful
consideration of the objections offered fails entirely to weaken the
argument. Example after example of a nature similar to those just quoted
might be given, indicating on the part of animals a perception of the
difference between 1 and 2, or between 2 and 3 and 4; and any reasoning
which tends to show that it is quantity rather than number which the animal
perceives, will apply with equal force to the Demara, the Chiquito, and the
Australian. Hence the actual origin of number may safely be excluded from
the limits of investigation, and, for the present, be left in the field of
pure speculation.

A most inviting field for research is, however, furnished by the primitive
methods of counting and of giving visible expression to the idea of number.
Our starting-point must, of course, be the sign language, which always
precedes intelligible speech; and which is so convenient and so expressive
a method of communication that the human family, even in its most highly
developed branches, never wholly lays it aside. It may, indeed, be stated
as a universal law, that some practical method of numeration has, in the
childhood of every nation or tribe, preceded the formation of numeral
words.

Practical methods of numeration are many in number and diverse in kind. But
the one primitive method of counting which seems to have been almost
universal throughout all time is the finger method. It is a matter of
common experience and observation that every child, when he begins to
count, turns instinctively to his fingers; and, with these convenient aids
as counters, tallies off the little number he has in mind. This method is
at once so natural and obvious that there can be no doubt that it has
always been employed by savage tribes, since the first appearance of the
human race in remote antiquity. All research among uncivilized peoples has
tended to confirm this view, were confirmation needed of anything so
patent. Occasionally some exception to this rule is found; or some
variation, such as is presented by the forest tribes of Brazil, who,
instead of counting on the fingers themselves, count on the joints of their
fingers.[5] As the entire number system of these tribes appears to be
limited to _three_, this variation is no cause for surprise.

The variety in practical methods of numeration observed among savage races,
and among civilized peoples as well, is so great that any detailed account
of them would be almost impossible. In one region we find sticks or splints
used; in another, pebbles or shells; in another, simple scratches, or
notches cut in a stick, Robinson Crusoe fashion; in another, kernels or
little heaps of grain; in another, knots on a string; and so on, in
diversity of method almost endless. Such are the devices which have been,
and still are, to be found in the daily habit of great numbers of Indian,
negro, Mongolian, and Malay tribes; while, to pass at a single step to the
other extremity of intellectual development, the German student keeps his
beer score by chalk marks on the table or on the wall. But back of all
these devices, and forming a common origin to which all may be referred, is
the universal finger method; the method with which all begin, and which all
find too convenient ever to relinquish entirely, even though their
civilization be of the highest type. Any such mode of counting, whether
involving the use of the fingers or not, is to be regarded simply as an
extraneous aid in the expression or comprehension of an idea which the mind
cannot grasp, or cannot retain, without assistance. The German student
scores his reckoning with chalk marks because he might otherwise forget;
while the Andaman Islander counts on his fingers because he has no other
method of counting,--or, in other words, of grasping the idea of number. A
single illustration may be given which typifies all practical methods of
numeration. More than a century ago travellers in Madagascar observed a
curious but simple mode of ascertaining the number of soldiers in an
army.[6] Each soldier was made to go through a passage in the presence of
the principal chiefs; and as he went through, a pebble was dropped on the
ground. This continued until a heap of 10 was obtained, when one was set
aside and a new heap begun. Upon the completion of 10 heaps, a pebble was
set aside to indicate 100; and so on until the entire army had been
numbered. Another illustration, taken from the very antipodes of
Madagascar, recently found its way into print in an incidental manner,[7]
and is so good that it deserves a place beside de Flacourt's time-honoured
example. Mom Cely, a Southern negro of unknown age, finds herself in debt
to the storekeeper; and, unwilling to believe that the amount is as great
as he represents, she proceeds to investigate the matter in her own
peculiar way. She had "kept a tally of these purchases by means of a
string, in which she tied commemorative knots." When her creditor
"undertook to make the matter clear to Cely's comprehension, he had to
proceed upon a system of her own devising. A small notch was cut in a
smooth white stick for every dime she owed, and a large notch when the
dimes amounted to a dollar; for every five dollars a string was tied in the
fifth big notch, Cely keeping tally by the knots in her bit of twine; thus,
when two strings were tied about the stick, the ten dollars were seen to be
an indisputable fact." This interesting method of computing the amount of
her debt, whether an invention of her own or a survival of the African life
of her parents, served the old negro woman's purpose perfectly; and it
illustrates, as well as a score of examples could, the methods of
numeration to which the children of barbarism resort when any number is to
be expressed which exceeds the number of counters with which nature has
provided them. The fingers are, however, often employed in counting numbers
far above the first decade. After giving the Il-Oigob numerals up to 60,
Müller adds:[8] "Above 60 all numbers, indicated by the proper figure
pantomime, are expressed by means of the word _ipi_." We know, moreover,
that many of the American Indian tribes count one ten after another on
their fingers; so that, whatever number they are endeavouring to indicate,
we need feel no surprise if the savage continues to use his fingers
throughout the entire extent of his counts. In rare instances we find
tribes which, like the Mairassis of the interior of New Guinea, appear to
use nothing but finger pantomime.[9] This tribe, though by no means
destitute of the number sense, is said to have no numerals whatever, but to
use the single word _awari_ with each show of fingers, no matter how few or
how many are displayed.

In the methods of finger counting employed by savages a considerable degree
of uniformity has been observed. Not only does he use his fingers to assist
him in his tally, but he almost always begins with the little finger of his
left hand, thence proceeding towards the thumb, which is 5. From this point
onward the method varies. Sometimes the second 5 also is told off on the
left hand, the same order being observed as in the first 5; but oftener the
fingers of the right hand are used, with a reversal of the order previously
employed; _i.e._ the thumb denotes 6, the index finger 7, and so on to the
little finger, which completes the count to 10.

At first thought there would seem to be no good reason for any marked
uniformity of method in finger counting. Observation among children fails
to detect any such thing; the child beginning, with almost entire
indifference, on the thumb or on the little finger of the left hand. My own
observation leads to the conclusion that very young children have a slight,
though not decided preference for beginning with the thumb. Experiments in
five different primary rooms in the public schools of Worcester, Mass.,
showed that out of a total of 206 children, 57 began with the little finger
and 149 with the thumb. But the fact that nearly three-fourths of the
children began with the thumb, and but one-fourth with the little finger,
is really far less significant than would appear at first thought. Children
of this age, four to eight years, will count in either way, and sometimes
seem at a loss themselves to know where to begin. In one school room where
this experiment was tried the teacher incautiously asked one child to count
on his fingers, while all the other children in the room watched eagerly to
see what he would do. He began with the little finger--and so did every
child in the room after him. In another case the same error was made by the
teacher, and the child first asked began with the thumb. Every other child
in the room did the same, each following, consciously or unconsciously, the
example of the leader. The results from these two schools were of course
rejected from the totals which are given above; but they serve an excellent
purpose in showing how slight is the preference which very young children
have in this particular. So slight is it that no definite law can be
postulated of this age; but the tendency seems to be to hold the palm of
the hand downward, and then begin with the thumb. The writer once saw a boy
about seven years old trying to multiply 3 by 6; and his method of
procedure was as follows: holding his left hand with its palm down, he
touched with the forefinger of his right hand the thumb, forefinger, and
middle finger successively of his left hand. Then returning to his
starting-point, he told off a second three in the same manner. This process
he continued until he had obtained 6 threes, and then he announced his
result correctly. If he had been a few years older, he might not have
turned so readily to his thumb as a starting-point for any digital count.
The indifference manifested by very young children gradually disappears,
and at the age of twelve or thirteen the tendency is decidedly in the
direction of beginning with the little finger. Fully three-fourths of all
persons above that age will be found to count from the little finger toward
the thumb, thus reversing the proportion that was found to obtain in the
primary school rooms examined.

With respect to finger counting among civilized peoples, we fail, then, to
find any universal law; the most that can be said is that more begin with
the little finger than with the thumb. But when we proceed to the study of
this slight but important particular among savages, we find them employing
a certain order of succession with such substantial uniformity that the
conclusion is inevitable that there must lie back of this some well-defined
reason, or perhaps instinct, which guides them in their choice. This
instinct is undoubtedly the outgrowth of the almost universal
right-handedness of the human race. In finger counting, whether among
children or adults, the beginning is made on the left hand, except in the
case of left-handed individuals; and even then the start is almost as
likely to be on the left hand as on the right. Savage tribes, as might be
expected, begin with the left hand. Not only is this custom almost
invariable, when tribes as a whole are considered, but the little finger is
nearly always called into requisition first. To account for this
uniformity, Lieutenant Gushing gives the following theory,[10] which is
well considered, and is based on the results of careful study and
observation among the Zuñi Indians of the Southwest: "Primitive man when
abroad never lightly quit hold of his weapons. If he wanted to count, he
did as the Zuñi afield does to-day; he tucked his instrument under his left
arm, thus constraining the latter, but leaving the right hand free, that he
might check off with it the fingers of the rigidly elevated left hand. From
the nature of this position, however, the palm of the left hand was
presented to the face of the counter, so that he had to begin his score on
the little finger of it, and continue his counting from the right leftward.
An inheritance of this may be detected to-day in the confirmed habit the
Zuñi has of gesticulating from the right leftward, with the fingers of the
right hand over those of the left, whether he be counting and summing up,
or relating in any orderly manner." Here, then, is the reason for this
otherwise unaccountable phenomenon. If savage man is universally
right-handed, he will almost inevitably use the index finger of his right
hand to mark the fingers counted, and he will begin his count just where it
is most convenient. In his case it is with the little finger of the left
hand. In the case of the child trying to multiply 3 by 6, it was with the
thumb of the same hand. He had nothing to tuck under his arm; so, in
raising his left hand to a position where both eye and counting finger
could readily run over its fingers, he held the palm turned away from his
face. The same choice of starting-point then followed as with the
savage--the finger nearest his right hand; only in this case the finger was
a thumb. The deaf mute is sometimes taught in this manner, which is for him
an entirely natural manner. A left-handed child might be expected to count
in a left-to-right manner, beginning, probably, with the thumb of his right
hand.

To the law just given, that savages begin to count on the little finger of
the left hand, there have been a few exceptions noted; and it has been
observed that the method of progression on the second hand is by no means
as invariable as on the first. The Otomacs[11] 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
Andamans[13] 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 Africa[15] the little finger of the right hand
is used for 1, and their count proceeds from right to left. With them, 6 is
the thumb of the left hand, 7 the forefinger, and so on. They hold the palm
downward instead of upward, and thus form a complete and striking exception
to the law which has been found to obtain with such substantial uniformity
in other parts of the uncivilized world. In Melanesia a few examples of
preference for beginning with the thumb may also be noticed. In the Banks
Islands the natives begin by turning down the thumb of the right hand, and
then the fingers in succession to the little finger, which is 5. This is
followed by the fingers of the left hand, both hands with closed fists
being held up to show the completed 10. In Lepers' Island, they begin with
the thumb, but, having reached 5 with the little finger, they do not pass
to the other hand, but throw up the fingers they have turned down,
beginning with the forefinger and keeping the thumb for 10.[16] In the use
of the single hand this people is quite peculiar. The second 5 is almost
invariably told off by savage tribes on the second hand, though in passing
from the one to the other primitive man does not follow any invariable law.
He marks 6 with either the thumb or the little finger. Probably the former
is the more common practice, but the statement cannot be made with any
degree of certainty. Among the Zulus the sequence is from thumb to thumb,
as is the case among the other South African tribes just mentioned; while
the Veis and numerous other African tribes pass from thumb to little
finger. The Eskimo, and nearly all the American Indian tribes, use the
correspondence between 6 and the thumb; but this habit is by no means
universal. Respecting progression from right to left or left to right on
the toes, there is no general law with which the author is familiar. Many
tribes never use the toes in counting, but signify the close of the first
10 by clapping the hands together, by a wave of the right hand, or by
designating some object; after which the fingers are again used as before.

One other detail in finger counting is worthy of a moment's notice. It
seems to have been the opinion of earlier investigators that in his passage
from one finger to the next, the savage would invariably bend down, or
close, the last finger used; that is, that the count began with the fingers
open and outspread. This opinion is, however, erroneous. Several of the
Indian tribes of the West[17] begin with the hand clenched, and open the
fingers one by one as they proceed. This method is much less common than
the other, but that it exists is beyond question.

In the Muralug Island, in the western part of Torres Strait, a somewhat
remarkable method of counting formerly existed, which grew out of, and is
to be regarded as an extension of, the digital method. Beginning with the
little finger of the left hand, the natives counted up to 5 in the usual
manner, and then, instead of passing to the other hand, or repeating the
count on the same fingers, they expressed the numbers from 6 to 10 by
touching and naming successively the left wrist, left elbow, left shoulder,
left breast, and sternum. Then the numbers from 11 to 19 were indicated by
the use, in inverse order, of the corresponding portions of the right side,
arm, and hand, the little finger of the right hand signifying 19. The words
used were in each case the actual names of the parts touched; the same
word, for example, standing for 6 and 14; but they were never used in the
numerical sense unless accompanied by the proper gesture, and bear no
resemblance to the common numerals, which are but few in number. This
method of counting is rapidly dying out among the natives of the island,
and is at the present time used only by old people.[18] Variations on this
most unusual custom have been found to exist in others of the neighbouring
islands, but none were exactly similar to it. One is also reminded by it of
a custom[19] which has for centuries prevailed among bargainers in the
East, of signifying numbers by touching the joints of each other's fingers
under a cloth. Every joint has a special signification; and the entire
system is undoubtedly a development from finger counting. The buyer or
seller will by this method express 6 or 60 by stretching out the thumb and
little finger and closing the rest of the fingers. The addition of the
fourth finger to the two thus used signifies 7 or 70; and so on. "It is
said that between two brokers settling a price by thus snipping with the
fingers, cleverness in bargaining, offering a little more, hesitating,
expressing an obstinate refusal to go further, etc., are as clearly
indicated as though the bargaining were being carried on in words.

The place occupied, in the intellectual development of man, by finger
counting and by the many other artificial methods of reckoning,--pebbles,
shells, knots, the abacus, etc.,--seems to be this: The abstract processes
of addition, subtraction, multiplication, division, and even counting
itself, present to the mind a certain degree of difficulty. To assist in
overcoming that difficulty, these artificial aids are called in; and, among
savages of a low degree of development, like the Australians, they make
counting possible. A little higher in the intellectual scale, among the
American Indians, for example, they are employed merely as an artificial
aid to what could be done by mental effort alone. Finally, among
semi-civilized and civilized peoples, the same processes are retained, and
form a part of the daily life of almost every person who has to do with
counting, reckoning, or keeping tally in any manner whatever. They are no
longer necessary, but they are so convenient and so useful that
civilization can never dispense with them. The use of the abacus, in the
form of the ordinary numeral frame, has increased greatly within the past
few years; and the time may come when the abacus in its proper form will
again find in civilized countries a use as common as that of five centuries
ago.

In the elaborate calculating machines of the present, such as are used by
life insurance actuaries and others having difficult computations to make,
we have the extreme of development in the direction of artificial aid to
reckoning. But instead of appearing merely as an extraneous aid to a
defective intelligence, it now presents itself as a machine so complex that
a high degree of intellectual power is required for the mere grasp of its
construction and method of working.





CHAPTER II.

NUMBER SYSTEM LIMITS.


With respect to the limits to which the number systems of the various
uncivilized races of the earth extend, recent anthropological research has
developed many interesting facts. In the case of the Chiquitos and a few
other native races of Bolivia we found no distinct number sense at all, as
far as could be judged from the absence, in their language, of numerals in
the proper sense of the word. How they indicated any number greater than
_one_ is a point still requiring investigation. In all other known
instances we find actual number systems, or what may for the sake of
uniformity be dignified by that name. In many cases, however, the numerals
existing are so few, and the ability to count is so limited, that the term
_number system_ is really an entire misnomer.

Among the rudest tribes, those whose mode of living approaches most nearly
to utter savagery, we find a certain uniformity of method. The entire
number system may consist of but two words, _one_ and _many_; or of three
words, _one_, _two_, _many_. Or, the count may proceed to 3, 4, 5, 10, 20,
or 100; passing always, or almost always, from the distinct numeral limit
to the indefinite _many_ or several, which serves for the expression of any
number not readily grasped by the mind. As a matter of fact, most races
count as high as 10; but to this statement the exceptions are so numerous
that they deserve examination in some detail. In certain parts of the
world, notably among the native races of South America, Australia, and many
of the islands of Polynesia and Melanesia, a surprising paucity of numeral
words has been observed. The Encabellada of the Rio Napo have but two
distinct numerals; _tey_, 1, and _cayapa_, 2.[20] The Chaco languages[21]
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 Puris[22] count _omi_, 1, _curiri_, 2, _prica_, many; and
the Botocudos[23] _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 Peru[25]
possess only three separate words for the expression of number,--_patrio_,
1, _pitteni_, 2, _mahuani_, 3. Above 3 they proceed by combinations, as 1
and 3 for 4, 1 and 1 and 3 for 5. Counting above 10 is, however, entirely
inconceivable to them, and any number beyond that limit they indicate by
_tohaine_, many. The Conibos,[26] of the same region, had, before their
contact with the Spanish, only _atchoupre_, 1, and _rrabui_, 2; though they
made some slight progress above 2 by means of reduplication. The Orejones,
one of the low, degraded tribes of the Upper Amazon,[27] have no names for
number except _nayhay_, 1, _nenacome_, 2, _feninichacome_, 3,
_ononoeomere_, 4. In the extensive vocabularies given by Von Martins,[28]
many similar examples are found. For the Bororos he gives only _couai_, 1,
_maeouai_, 2, _ouai_, 3. The last word, with the proper finger pantomime,
serves also for any higher number which falls within the grasp of their
comprehension. The Guachi manage to reach 5, but their numeration is of the
rudest kind, as the following scale shows: _tamak_, 1, _eu-echo,_ 2,
_eu-echo-kailau,_ 3, _eu-echo-way,_ 4, _localau_, 5. The Carajas counted by
a scale equally rude, and their conception of number seemed equally vague,
until contact with the neighbouring tribes furnished them with the means of
going beyond their original limit. Their scale shows clearly the uncertain,
feeble number sense which is so marked in the interior of South America. It
contains _wadewo_, 1, _wadebothoa_, 2, _wadeboaheodo_, 3, _wadebojeodo_,
4, _wadewajouclay_, 5, _wadewasori_, 6, or many.

Turning to the languages of the extinct, or fast vanishing, tribes of
Australia, we find a still more noteworthy absence of numeral expressions.
In the Gudang dialect[29] but two numerals are found--_pirman_, 1, and
_ilabiu_, 2; in the Weedookarry, _ekkamurda_, 1, and _kootera_, 2; and in
the Queanbeyan, _midjemban_, 1, and _bollan_, 2. In a score or more of
instances the numerals stop at 3. The natives of Keppel Bay count _webben_,
1, _booli_, 2, _koorel_, 3; of the Boyne River, _karroon_, 1, _boodla_, 2,
_numma_, 3; of the Flinders River, _kooroin_, 1, _kurto_, 2, _kurto
kooroin_, 3; at the mouth of the Norman River, _lum_, 1, _buggar_, 2,
_orinch_, 3; the Eaw tribe, _koothea_, 1, _woother_, 2, _marronoo_, 3; the
Moree, _mal_, 1, _boolar_, 2, _kooliba_, 3; the Port Essington,[30] _erad_,
1, _nargarick_, 2, _nargarickelerad_, 3; the Darnly Islanders,[31] _netat_,
1, _naes_, 2, _naesa netat_, 3; and so on through a long list of tribes
whose numeral scales are equally scanty. A still larger number of tribes
show an ability to count one step further, to 4; but beyond this limit the
majority of Australian and Tasmanian tribes do not go. It seems most
remarkable that any human being should possess the ability to count to 4,
and not to 5. The number of fingers on one hand furnishes so obvious a
limit to any of these rudimentary systems, that positive evidence is needed
before one can accept the statement. A careful examination of the numerals
in upwards of a hundred Australian dialects leaves no doubt, however, that
such is the fact. The Australians in almost all cases count by pairs; and
so pronounced is this tendency that they pay but little attention to the
fingers. Some tribes do not appear ever to count beyond 2--a single pair.
Many more go one step further; but if they do, they are as likely as not to
designate their next numeral as two-one, or possibly, one-two. If this step
is taken, we may or may not find one more added to it, thus completing the
second pair. Still, the Australian's capacity for understanding anything
which pertains to number is so painfully limited that even here there is
sometimes an indefinite expression formed, as many, heap, or plenty,
instead of any distinct numeral; and it is probably true that no Australian
language contains a pure, simple numeral for 4. Curr, the best authority on
this subject, believes that, where a distinct word for 4 is given,
investigators have been deceived in every case.[32] If counting is carried
beyond 4, it is always by means of reduplication. A few tribes gave
expressions for 5, fewer still for 6, and a very small number appeared able
to reach 7. Possibly the ability to count extended still further; but if
so, it consisted undoubtedly in reckoning one pair after another, without
any consciousness whatever of the sum total save as a larger number.

The numerals of a few additional tribes will show clearly that all distinct
perception of number is lost as soon as these races attempt to count above
3, or at most, 4. The Yuckaburra[33] natives can go no further than
_wigsin_, 1, _bullaroo_, 2, _goolbora_, 3. Above here all is referred to as
_moorgha_, many. The Marachowies[34] 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 dialect[35] 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
rivers[36] the scale appears as _yaburu_, 1, _boolaroo_, 2, _boolaroo
yaburu_, 3, and _gurul_ for 4 or anything beyond. The Wiraduroi[37] have
_numbai_, 1, _bula_, 2, _bula numbai_, 3, _bungu_, 4, or many, and _bungu
galan_ or _bian galan_, 5, or very many. The Kamilaroi[38] 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 Tasmanians[39]
were found to be unable to proceed beyond 2; and as the foregoing examples
have indicated, their Australian neighbours were but little better off. In
one or two instances we do find Australian numeral scales which reach 10,
and perhaps we may safely say 20. One of these is given in full in a
subsequent chapter, and its structure gives rise to the suspicion that it
was originally as limited as those of kindred tribes, and that it underwent
a considerable development after the natives had come in contact with the
Europeans. There is good reason to believe that no Australian in his wild
state could ever count intelligently to 7.[40]

In certain portions of Asia, Africa, Melanesia, Polynesia, and North
America, are to be found races whose number systems are almost and
sometimes quite as limited as are those of the South. American and
Australian tribes already cited, but nowhere else do we find these so
abundant as in the two continents just mentioned, where example after
example might be cited of tribes whose ability to count is circumscribed
within the narrowest limits. The Veddas[41] of Ceylon have but two
numerals, _ekkame[=i]_, 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 Bushmen[43] of South Africa have but two numerals, the pronunciation of
which can hardly be indicated without other resources than those of the
English alphabet. Their word for 3 means, simply, many, as in the case of
some of the Australian tribes. The Watchandies[44] 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 California[45]
cannot count above 5. A few of the more intelligent among them understand
the meaning of 2 fives, but this number seems entirely beyond the
comprehension of the ordinary native. The Comanches, curiously enough, are
so reluctant to employ their number words that they appear to prefer finger
pantomime instead, thus giving rise to the impression which at one time
became current, that they had no numerals at all for ordinary counting.

Aside from the specific examples already given, a considerable number of
sweeping generalizations may be made, tending to show how rudimentary the
number sense may be in aboriginal life. Scores of the native dialects of
Australia and South America have been found containing number systems but
little more extensive than those alluded to above. The negro tribes of
Africa give the same testimony, as do many of the native races of Central
America, Mexico, and the Pacific coast of the United States and Canada, the
northern part of Siberia, Greenland, Labrador, and the arctic archipelago.
In speaking of the Eskimos of Point Barrow, Murdoch[46] says: "It was not
easy to obtain any accurate information about the numeral system of these
people, since in ordinary conversation they are not in the habit of
specifying any numbers above five." Counting is often carried higher than
this among certain of these northern tribes, but, save for occasional
examples, it is limited at best. Dr. Franz Boas, who has travelled
extensively among the Eskimos, and whose observations are always of the
most accurate nature, once told the author that he never met an Eskimo who
could count above 15. Their numerals actually do extend much higher; and a
stray numeral of Danish origin is now and then met with, showing that the
more intelligent among them are able to comprehend numbers of much greater
magnitude than this. But as Dr. Boas was engaged in active work among them
for three years, we may conclude that the Eskimo has an arithmetic but
little more extended than that which sufficed for the Australians and the
forest tribes of Brazil. Early Russian explorers among the northern tribes
of Siberia noticed the same difficulty in ordinary, every-day reckoning
among the natives. At first thought we might, then, state it as a general
law that those races which are lowest in the scale of civilization, have
the feeblest number sense also; or in other words, the least possible power
of grasping the abstract idea of number.

But to this law there are many and important exceptions. The concurrent
testimony of explorers seems to be that savage races possess, in the great
majority of cases, the ability to count at least as high as 10. This limit
is often extended to 20, and not infrequently to 100. Again, we find 1000
as the limit; or perhaps 10,000; and sometimes the savage carries his
number system on into the hundreds of thousands or millions. Indeed, the
high limit to which some savage races carry their numeration is far more
worthy of remark than the entire absence of the number sense exhibited by
others of apparently equal intelligence. If the life of any tribe is such
as to induce trade and barter with their neighbours, a considerable
quickness in reckoning will be developed among them. Otherwise this power
will remain dormant because there is but little in the ordinary life of
primitive man to call for its exercise.

In giving 1, 2, 3, 5, 10, or any other small number as a system limit, it
must not be overlooked that this limit mentioned is in all cases the limit
of the spoken numerals at the savage's command. The actual ability to count
is almost always, and one is tempted to say always, somewhat greater than
their vocabularies would indicate. The Bushman has no number word that will
express for him anything higher than 2; but with the assistance of his
fingers he gropes his way on as far as 10. The Veddas, the Andamans, the
Guachi, the Botocudos, the Eskimos, and the thousand and one other tribes
which furnish such scanty numeral systems, almost all proceed with more or
less readiness as far as their fingers will carry them. As a matter of
fact, this limit is frequently extended to 20; the toes, the fingers of a
second man, or a recount of the savage's own fingers, serving as a tale for
the second 10. Allusion is again made to this in a later chapter, where the
subject of counting on the fingers and toes is examined more in detail.

In saying that a savage can count to 10, to 20, or to 100, but little idea
is given of his real mental conception of any except the smallest numbers.
Want of familiarity with the use of numbers, and lack of convenient means
of comparison, must result in extreme indefiniteness of mental conception
and almost entire absence of exactness. The experience of Captain
Parry,[47] who found that the Eskimos made mistakes before they reached 7,
and of Humboldt,[48] who says that a Chayma might be made to say that his
age was either 18 or 60, has been duplicated by all investigators who have
had actual experience among savage races. Nor, on the other hand, is the
development of a numeral system an infallible index of mental power, or of
any real approach toward civilization. A continued use of the trading and
bargaining faculties must and does result in a familiarity with numbers
sufficient to enable savages to perform unexpected feats in reckoning.
Among some of the West African tribes this has actually been found to be
the case; and among the Yorubas of Abeokuta[49] the extraordinary saying,
"You may seem very clever, but you can't tell nine times nine," shows how
surprisingly this faculty has been developed, considering the general
condition of savagery in which the tribe lived. There can be no doubt that,
in general, the growth of the number sense keeps pace with the growth of
the intelligence in other respects. But when it is remembered that the
Tonga Islanders have numerals up to 100,000, and the Tembus, the Fingoes,
the Pondos, and a dozen other South African tribes go as high as 1,000,000;
and that Leigh Hunt never could learn the multiplication table, one must
confess that this law occasionally presents to our consideration remarkable
exceptions.

While considering the extent of the savage's arithmetical knowledge, of his
ability to count and to grasp the meaning of number, it may not be amiss to
ask ourselves the question, what is the extent of the development of our
own number sense? To what limit can we absorb the idea of number, with a
complete appreciation of the idea of the number of units involved in any
written or spoken quantity? Our perfect system of numeration enables us to
express without difficulty any desired number, no matter how great or how
small it be. But how much of actually clear comprehension does the number
thus expressed convey to the mind? We say that one place is 100 miles from
another; that A paid B 1000 dollars for a certain piece of property; that a
given city contains 10,000 inhabitants; that 100,000 bushels of wheat were
shipped from Duluth or Odessa on such a day; that 1,000,000 feet of lumber
were destroyed by the fire of yesterday,--and as we pass from the smallest
to the largest of the numbers thus instanced, and from the largest on to
those still larger, we repeat the question just asked; and we repeat it
with a new sense of our own mental limitation. The number 100
unquestionably stands for a distinct conception. Perhaps the same may be
said for 1000, though this could not be postulated with equal certainty.
But what of 10,000? If that number of persons were gathered together into a
single hall or amphitheatre, could an estimate be made by the average
onlooker which would approximate with any degree of accuracy the size of
the assembly? Or if an observer were stationed at a certain point, and
10,000 persons were to pass him in single file without his counting them as
they passed, what sort of an estimate would he make of their number? The
truth seems to be that our mental conception of number is much more limited
than is commonly thought, and that we unconsciously adopt some new unit as
a standard of comparison when we wish to render intelligible to our minds
any number of considerable magnitude. For example, we say that A has a
fortune of $1,000,000. The impression is at once conveyed of a considerable
degree of wealth, but it is rather from the fact that that fortune
represents an annual income of $40,000 than, from the actual magnitude of
the fortune itself. The number 1,000,000 is, in itself, so greatly in
excess of anything that enters into our daily experience that we have but a
vague conception of it, except as something very great. We are not, after
all, so very much better off than the child who, with his arms about his
mother's neck, informs her with perfect gravity and sincerity that he
"loves her a million bushels." His idea is merely of some very great
amount, and our own is often but little clearer when we use the expressions
which are so easily represented by a few digits. Among the uneducated
portions of civilized communities the limit of clear comprehension of
number is not only relatively, but absolutely, very low. Travellers in
Russia have informed the writer that the peasants of that country have no
distinct idea of a number consisting of but a few hundred even. There is no
reason to doubt this testimony. The entire life of a peasant might be
passed without his ever having occasion to use a number as great as 500,
and as a result he might have respecting that number an idea less distinct
than a trained mathematician would have of the distance from the earth to
the sun. De Quincey[50] incidentally mentions this characteristic in
narrating a conversation which occurred while he was at Carnarvon, a little
town in Wales. "It was on this occasion," he says, "that I learned how
vague are the ideas of number in unpractised minds. 'What number of people
do you think,' I said to an elderly person, 'will be assembled this day at
Carnarvon?' 'What number?' rejoined the person addressed; 'what number?
Well, really, now, I should reckon--perhaps a matter of four million.' Four
millions of _extra_ people in little Carnarvon, that could barely find
accommodation (I should calculate) for an extra four hundred!" So the
Eskimo and the South American Indian are, after all, not so very far behind
the "elderly person" of Carnarvon, in the distinct perception of a number
which familiarity renders to us absurdly small.





CHAPTER III.

THE ORIGIN OF NUMBER WORDS.


In the comparison of languages and the search for primitive root forms, no
class of expressions has been subjected to closer scrutiny than the little
cluster of words, found in each language, which constitutes a part of the
daily vocabulary of almost every human being--the words with which we begin
our counting. It is assumed, and with good reason, that these are among the
earlier words to appear in any language; and in the mutations of human
speech, they are found to suffer less than almost any other portion of a
language. Kinship between tongues remote from each other has in many
instances been detected by the similarity found to exist among the
every-day words of each; and among these words one may look with a good
degree of certainty for the 1, 2, 3, etc., of the number scale. So fruitful
has been this line of research, that the attempt has been made, even, to
establish a common origin for all the races of mankind by means of a
comparison of numeral words.[51] But in this instance, as in so many others
that will readily occur to the mind, the result has been that the theory
has finally taken possession of the author and reduced him to complete
subjugation, instead of remaining his servant and submitting to the
legitimate results of patient and careful investigation. Linguistic
research is so full of snares and pitfalls that the student must needs
employ the greatest degree of discrimination before asserting kinship of
race because of resemblances in vocabulary; or even relationship between
words in the same language because of some chance likeness of form that may
exist between them. Probably no one would argue that the English and the
Babusessé of Central Africa were of the same primitive stock simply because
in the language of the latter _five atano_ means 5, and _ten kumi_ means
10.[52] But, on the other hand, many will argue that, because the German
_zehn_ means 10, and _zehen_ means toes, the ancestors of the Germans
counted on their toes; and that with them, 10 was the complete count of the
toes. It may be so. We certainly have no evidence with which to disprove
this; but, before accepting it as a fact, or even as a reasonable
hypothesis, we may be pardoned for demanding some evidence aside from the
mere resemblance in the form of the words. If, in the study of numeral
words, form is to constitute our chief guide, we must expect now and then
to be confronted with facts which are not easily reconciled with any pet
theory.

The scope of the present work will admit of no more than a hasty
examination of numeral forms, in which only actual and well ascertained
meanings will be considered. But here we are at the outset confronted with
a class of words whose original meanings appear to be entirely lost. They
are what may be termed the numerals proper--the native, uncompounded words
used to signify number. Such words are the one, two, three, etc., of
English; the eins, zwei, drei, etc., of German; words which must at some
time, in some prehistoric language, have had definite meanings entirely
apart from those which they now convey to our minds. In savage languages it
is sometimes possible to detect these meanings, and thus to obtain
possession of the clue that leads to the development, in the barbarian's
rude mind, of a count scale--a number system. But in languages like those
of modern Europe, the pedigree claimed by numerals is so long that, in the
successive changes through which they have passed, all trace of their
origin seems to have been lost.

The actual number of such words is, however, surprisingly small in any
language. In English we count by simple words only to 10. From this point
onward all our numerals except "hundred" and "thousand" are compounds and
combinations of the names of smaller numbers. The words we employ to
designate the higher orders of units, as million, billion, trillion, etc.,
are appropriated bodily from the Italian; and the native words _pair_,
_tale_, _brace_, _dozen_, _gross_, and _score_, can hardly be classed as
numerals in the strict sense of the word. German possesses exactly the same
number of native words in its numeral scale as English; and the same may be
said of the Teutonic languages generally, as well as of the Celtic, the
Latin, the Slavonic, and the Basque. This is, in fact, the universal method
observed in the formation of any numeral scale, though the actual number of
simple words may vary. The Chiquito language has but one numeral of any
kind whatever; English contains twelve simple terms; Sanskrit has
twenty-seven, while Japanese possesses twenty-four, and the Chinese a
number almost equally great. Very many languages, as might be expected,
contain special numeral expressions, such as the German _dutzend_ and the
French _dizaine_; but these, like the English _dozen_ and _score_, are not
to be regarded as numerals proper.

The formation of numeral words shows at a glance the general method in
which any number scale has been built up. The primitive savage counts on
his fingers until he has reached the end of one, or more probably of both,
hands. Then, if he wishes to proceed farther, some mark is made, a pebble
is laid aside, a knot tied, or some similar device employed to signify that
all the counters at his disposal have been used. Then the count begins
anew, and to avoid multiplication of words, as well as to assist the
memory, the terms already used are again resorted to; and the name by which
the first halting-place was designated is repeated with each new numeral.
Hence the thirteen, fourteen, fifteen, etc., which are contractions of the
fuller expressions three-and-ten, four-and-ten, five-and-ten, etc. The
specific method of combination may not always be the same, as witness the
_eighteen_, or eight-ten, in English, and _dix-huit,_ or ten-eight, in
French; _forty-five_, or four-tens-five, in English, and _fünf und
vierzig_, or five and four tens in German. But the general method is the
same the world over, presenting us with nothing but local variations, which
are, relatively speaking, entirely unimportant. With this fact in mind, we
can cease to wonder at the small number of simple numerals in any language.
It might, indeed, be queried, why do any languages, English and German, for
example, have unusual compounds for 11 and 12? It would seem as though the
regular method of compounding should begin with 10 and 1, instead of 10 and
3, in any language using a system with 10 as a base. An examination of
several hundred numeral scales shows that the Teutonic languages are
somewhat exceptional in this respect. The words _eleven_ and _twelve_ are
undoubtedly combinations, but not in the same direct sense as _thirteen_,
_twenty-five_, etc. The same may be said of the French _onze_, _douze_,
_treize_, _quatorze_, _quinze_, and _seize_, which are obvious compounds,
but not formed in the same manner as the numerals above that point. Almost
all civilized languages, however, except the Teutonic, and practically all
uncivilized languages, begin their direct numeral combinations as soon as
they have passed their number base, whatever that may be. To give an
illustration, selected quite at random from among the barbarous tribes of
Africa, the Ki-Swahili numeral scale runs as follows:[53]

   1.  moyyi,
   2.  mbiri,
   3.  tato,
   4.  ena,
   5.  tano,
   6.  seta,
   7.  saba,
   8.  nani,
   9.  kenda,
  10.  kumi,
  11.  kumi na moyyi,
  12.  kumi na mbiri,
  13.  kumi na tato,
  etc.

The words for 11, 12, and 13, are seen at a glance to signify ten-and-one,
ten-and-two, ten-and-three, and the count proceeds, as might be inferred,
in a similar manner as far as the number system extends. Our English
combinations are a little closer than these, and the combinations found in
certain other languages are, in turn, closer than those of the English; as
witness the _once_, 11, _doce_, 12, _trece_, 13, etc., of Spanish. But the
process is essentially the same, and the law may be accepted as practically
invariable, that all numerals greater than the base of a system are
expressed by compound words, except such as are necessary to establish some
new order of unit, as hundred or thousand.

In the scale just given, it will be noticed that the larger number precedes
the smaller, giving 10 + 1, 10 + 2, etc., instead of 1 + 10, 2 + 10, etc.
This seems entirely natural, and hardly calls for any comment whatever. But
we have only to consider the formation of our English "teens" to see that
our own method is, at its inception, just the reverse of this. Thirteen,
14, and the remaining numerals up to 19 are formed by prefixing the smaller
number to the base; and it is only when we pass 20 that we return to the
more direct and obvious method of giving precedence to the larger. In
German and other Teutonic languages the inverse method is continued still
further. Here 25 is _fünf und zwanzig_, 5 and 20; 92 is _zwei und neunzig_,
2 and 90, and so on to 99. Above 100 the order is made direct, as in
English. Of course, this mode of formation between 20 and 100 is
permissible in English, where "five and twenty" is just as correct a form
as twenty-five. But it is archaic, and would soon pass out of the language
altogether, were it not for the influence of some of the older writings
which have had a strong influence in preserving for us many of older and
more essentially Saxon forms of expression.

Both the methods described above are found in all parts of the world, but
what I have called the direct is far more common than the other. In
general, where the smaller number precedes the larger it signifies
multiplication instead of addition. Thus, when we say "thirty," _i.e._
three-ten, we mean 3 × 10; just as "three hundred" means 3 × 100. When the
larger precedes the smaller, we must usually understand addition. But to
both these rules there are very many exceptions. Among higher numbers the
inverse order is very rarely used; though even here an occasional exception
is found. The Taensa Indians, for example, place the smaller numbers before
the larger, no matter how far their scale may extend. To say 1881 they make
a complete inversion of our own order, beginning with 1 and ending with
1000. Their full numeral for this is _yeha av wabki mar-u-wab mar-u-haki_,
which means, literally, 1 + 80 + 100 × 8 + 100 × 10.[54] Such exceptions
are, however, quite rare.

One other method of combination, that of subtraction, remains to be
considered. Every student of Latin will recall at once the _duodeviginti_,
2 from 20, and _undeviginti_, 1 from 20, which in that language are the
regular forms of expression for 18 and 19. At first they seem decidedly
odd; but familiarity soon accustoms one to them, and they cease entirely to
attract any special attention. This principle of subtraction, which, in the
formation of numeral words, is quite foreign to the genius of English, is
still of such common occurrence in other languages that the Latin examples
just given cease to be solitary instances.

The origin of numerals of this class is to be found in the idea of
reference, not necessarily to the last, but to the nearest, halting-point
in the scale. Many tribes seem to regard 9 as "almost 10," and to give it a
name which conveys this thought. In the Mississaga, one of the numerous
Algonquin languages, we have, for example, the word _cangaswi_, "incomplete
10," for 9.[55] In the Kwakiutl of British Columbia, 8 as well as 9 is
formed in this way; these two numbers being _matlguanatl_, 10 - 2, and
_nanema_, 10 - 1, respectively.[56] In many of the languages of British
Columbia we find a similar formation for 8 and 9, or for 9 alone. The same
formation occurs in Malay, resulting in the numerals _delapan_, 10 - 2, and
_sambilan_ 10 - 1.[57] In Green Island, one of the New Ireland group, these
become simply _andra-lua_, "less 2," and _andra-si_, "less 1."[58] In the
Admiralty Islands this formation is carried back one step further, and not
only gives us _shua-luea_, "less 2," and _shu-ri_, "less 1," but also makes
7 appear as _sua-tolu_, "less 3."[59] Surprising as this numeral is, it is
more than matched by the Ainu scale, which carries subtraction back still
another step, and calls 6, 10 - 4. The four numerals from 6 to 9 in this
scale are respectively, _iwa_, 10 - 4, _arawa_, 10 - 3, _tupe-san_, 10 - 2,
and _sinepe-san_, 10 - 1.[60] Numerous examples of this kind of formation
will be found in later chapters of this work; but they will usually be
found to occur in one or both of the numerals, 8 and 9. Occasionally they
appear among the higher numbers; as in the Maya languages, where, for
example, 99 years is "one single year lacking from five score years,"[61]
and in the Arikara dialects, where 98 and 99 are "5 men minus" and "5 men 1
not."[62] The Welsh, Danish, and other languages less easily accessible
than these to the general student, also furnish interesting examples of a
similar character.

More rarely yet are instances met with of languages which make use of
subtraction almost as freely as addition, in the composition of numerals.
Within the past few years such an instance has been noticed in the case of
the Bellacoola language of British Columbia. In their numeral scale 15,
"one foot," is followed by 16, "one man less 4"; 17, "one man less 3"; 18,
"one man less 2"; 19, "one man less 1"; and 20, one man. Twenty-five is
"one man and one hand"; 26, "one man and two hands less 4"; 36, "two men
less 4"; and so on. This method of formation prevails throughout the entire
numeral scale.[63]

One of the best known and most interesting examples of subtraction as
a well-defined principle of formation is found in the Maya scale. Up
to 40 no special peculiarity appears; but as the count progresses beyond
that point we find a succession of numerals which one is almost tempted
to call 60 - 19, 60 - 18, 60 - 17, etc. Literally translated the meanings
seem to be 1 to 60, 2 to 60, 3 to 60, etc. The point of reference is 60,
and the thought underlying the words may probably be expressed by the
paraphrases, "1 on the third score, 2 on the third score, 3 on the third
score," etc.  Similarly, 61 is 1 on the fourth score, 81 is one on the
fifth score, 381 is 1 on the nineteenth score, and so on to 400. At 441
the same formation reappears; and it continues to characterize the system
in a regular and consistent manner, no matter how far it is extended.[64]

The Yoruba language of Africa is another example of most lavish use of
subtraction; but it here results in a system much less consistent and
natural than that just considered. Here we find not only 5, 10, and 20
subtracted from the next higher unit, but also 40, and even 100. For
example, 360 is 400 - 40; 460 is 500 - 40; 500 is 600 - 100; 1300 is
1400 - 100, etc. One of the Yoruba units is 200; and all the odd hundreds
up to 2000, the next higher unit, are formed by subtracting 100 from the
next higher multiple of 200. The system is quite complex, and very
artificial; and seems to have been developed by intercourse with
traders.[65]

It has already been stated that the primitive meanings of our own simple
numerals have been lost. This is also true of the languages of nearly all
other civilized peoples, and of numerous savage races as well. We are at
liberty to suppose, and we do suppose, that in very many cases these words
once expressed meanings closely connected with the names of the fingers, or
with the fingers themselves, or both. Now and then a case is met with in
which the numeral word frankly avows its meaning--as in the Botocudo
language, where 1 is expressed by _podzik_, finger, and 2 by _kripo_,
double finger;[66] and in the Eskimo dialect of Hudson's Bay, where
_eerkitkoka_ means both 10 and little finger.[67] Such cases are, however,
somewhat exceptional.

In a few noteworthy instances, the words composing the numeral scale of a
language have been carefully investigated and their original meanings
accurately determined. The simple structure of many of the rude languages
of the world should render this possible in a multitude of cases; but
investigators are too often content with the mere numerals themselves, and
make no inquiry respecting their meanings. But the following exposition of
the Zuñi scale, given by Lieutenant Gushing[68] leaves nothing to be
desired:

   1.  töpinte = taken to start with.
   2.  kwilli  = put down together with.
   3.  ha'[=i] = the equally dividing finger.
   4.  awite   = all the fingers all but done with.
   5.  öpte    = the notched off.

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

   6.  topalïk'ya   = another brought to add to the done with.
   7.  kwillilïk'ya = two brought to and held up with the rest.
   8.  hailïk'ye    = three brought to and held up with the rest.
   9.  tenalïk'ya   = all but all are held up with the rest.
  10.  ästem'thila  = all the fingers.
  11.  ästem'thla topayä'thl'tona = all the fingers and another over
                                    above held.

The process of formation indicated in 11 is used in the succeeding numerals
up to 19.

    20.  kwillik'yënästem'thlan = two times all the fingers.
   100.  ässiästem'thlak'ya = the fingers all the fingers.
  1000.  ässiästem'thlanak'yënästem'thla = the fingers all the fingers
                                           times all the fingers.

The only numerals calling for any special note are those for 11 and 9. For
9 we should naturally expect a word corresponding in structure and meaning
to the words for 7 and 8. But instead of the "four brought to and held up
with the rest," for which we naturally look, the Zuñi, to show that he has
used all of his fingers but one, says "all but all are held up with the
rest." To express 11 he cannot use a similar form of composition, since he
has already used it in constructing his word for 6, so he says "all the
fingers and another over above held."

The one remarkable point to be noted about the Zuñi scale is, after all,
the formation of the words for 1 and 2. While the savage almost always
counts on his fingers, it does not seem at all certain that these words
would necessarily be of finger formation. The savage can always distinguish
between one object and two objects, and it is hardly reasonable to believe
that any external aid is needed to arrive at a distinct perception of this
difference. The numerals for 1 and 2 would be the earliest to be formed in
any language, and in most, if not all, cases they would be formed long
before the need would be felt for terms to describe any higher number. If
this theory be correct, we should expect to find finger names for numerals
beginning not lower than 3, and oftener with 5 than with any other number.
The highest authority has ventured the assertion that all numeral words
have their origin in the names of the fingers;[69] substantially the same
conclusion was reached by Professor Pott, of Halle, whose work on numeral
nomenclature led him deeply into the study of the origin of these words.
But we have abundant evidence at hand to show that, universal as finger
counting has been, finger origin for numeral words has by no means been
universal. That it is more frequently met with than any other origin is
unquestionably true; but in many instances, which will be more fully
considered in the following chapter, we find strictly non-digital
derivations, especially in the case of the lowest members of the scale. But
in nearly all languages the origin of the words for 1, 2, 3, and 4 are so
entirely unknown that speculation respecting them is almost useless.

An excellent illustration of the ordinary method of formation which obtains
among number scales is furnished by the Eskimos of Point Barrow,[70] who
have pure numeral words up to 5, and then begin a systematic course of word
formation from the names of their fingers. If the names of the first five
numerals are of finger origin, they have so completely lost their original
form, or else the names of the fingers themselves have so changed, that no
resemblance is now to be detected between them. This scale is so
interesting that it is given with considerable fulness, as follows:

   1.  atauzik.
   2.  madro.
   3.  pinasun.
   4.  sisaman.
   5.  tudlemut.
   6.  atautyimin akbinigin [tudlimu(t)] = 5 and 1 on the next.
   7.  madronin akbinigin  = twice on the next.
   8.  pinasunin akbinigin = three times on the next.
   9.  kodlinotaila        = that which has not its 10.
  10.  kodlin              = the upper part--_i.e._ the fingers.
  14.  akimiaxotaityuna    = I have not 15.
  15.  akimia. [This seems to be a real numeral word.]
  20.  inyuina = a man come to an end.
  25.  inyuina tudlimunin akbinidigin = a man come to an end and 5 on the
                                        next.
  30.  inyuina kodlinin akbinidigin = a man come to an end and 10 on the
                                      next.
  35.  inyuina akimiamin aipalin = a man come to an end accompanied by 1
                                   fifteen times.
  40.  madro inyuina = 2 men come to an end.

In this scale we find the finger origin appearing so clearly and so
repeatedly that one feels some degree of surprise at finding 5 expressed by
a pure numeral instead of by some word meaning _hand_ or _fingers of one
hand_. In this respect the Eskimo dialects are somewhat exceptional among
scales built up of digital words. The system of the Greenland Eskimos,
though differing slightly from that of their Point Barrow cousins, shows
the same peculiarity. The first ten numerals of this scale are:[71]

   1.  atausek.
   2.  mardluk.
   3.  pingasut.
   4.  sisamat.
   5.  tatdlimat.
   6.  arfinek-atausek  = to the other hand 1.
   7.  arfinek-mardluk  = to the other hand 2.
   8.  arfinek-pingasut = to the other hand 3.
   9.  arfinek-sisamat  = to the other hand 4.
  10.  kulit.

The same process is now repeated, only the feet instead of the hands are
used; and the completion of the second 10 is marked by the word _innuk_,
man. It may be that the Eskimo word for 5 is, originally, a digital word,
but if so, the fact has not yet been detected. From the analogy furnished
by other languages we are justified in suspecting that this may be the
case; for whenever a number system contains digital words, we expect them
to begin with _five_, as, for example, in the Arawak scale,[72] which runs:

   1.  abba.
   2.  biama.
   3.  kabbuhin.
   4.  bibiti.
   5.  abbatekkábe = 1 hand.
   6.  abbatiman   = 1 of the other.
   7.  biamattiman = 2 of the other.
   8.  kabbuhintiman = 3 of the other.
   9.  bibitiman     = 4 of the other.
  10.  biamantekábbe = 2 hands.
  11.  abba kutihibena = 1 from the feet.
  20.  abba lukku = hands feet.

The four sets of numerals just given may be regarded as typifying one of
the most common forms of primitive counting; and the words they contain
serve as illustrations of the means which go to make up the number scales
of savage races. Frequently the finger and toe origin of numerals is
perfectly apparent, as in the Arawak system just given, which exhibits the
simplest and clearest possible method of formation. Another even more
interesting system is that of the Montagnais of northern Canada.[73] Here,
as in the Zuñi scale, the words are digital from the outset.

   1.  inl'are = the end is bent.
   2.  nak'e   = another is bent.
   3.  t'are   = the middle is bent.
   4.  dinri   = there are no more except this.
   5.  se-sunla-re = the row on the hand.
   6.  elkke-t'are = 3 from each side.
   7.{ t'a-ye-oyertan = there are still 3 of them.
     { inl'as dinri   = on one side there are 4 of them.
   8.  elkke-dinri    = 4 on each side.
   9.  inl'a-ye-oyert'an = there is still 1 more.
  10.  onernan = finished on each side.
  11.  onernan inl'are ttcharidhel = 1 complete and 1.
  12.  onernan nak'e ttcharidhel = 1 complete and 2, etc.

The formation of 6, 7, and 8 of this scale is somewhat different from that
ordinarily found. To express 6, the Montagnais separates the thumb and
forefinger from the three remaining fingers of the left hand, and bringing
the thumb of the right hand close to them, says: "3 from each side." For 7
he either subtracts from 10, saying: "there are still 3 of them," or he
brings the thumb and forefinger of the right hand up to the thumb of the
left, and says: "on one side there are 4 of them." He calls 8 by the same
name as many of the other Canadian tribes, that is, two 4's; and to show
the proper number of fingers, he closes the thumb and little finger of the
right hand, and then puts the three remaining fingers beside the thumb of
the left hand. This method is, in some of these particulars, different from
any other I have ever examined.

It often happens that the composition of numeral words is less easily
understood, and the original meanings more difficult to recover, than in
the examples already given. But in searching for number systems which show
in the formation of their words the influence of finger counting, it is not
unusual to find those in which the derivation from native words signifying
_finger, hand, toe, foot_, and _man_, is just as frankly obvious as in the
case of the Zuñi, the Arawak, the Eskimo, or the Montagnais scale. Among
the Tamanacs,[74] one of the numerous Indian tribes of the Orinoco, the
numerals are as strictly digital as in any of the systems already examined.
The general structure of the Tamanac scale is shown by the following
numerals:

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

In the Guarani[75] language of Paraguay the same method is found, with a
different form of expression for 20. Here the numerals in question are

   5.  asepopetei = one hand.
  10.  asepomokoi = two hands.
  20.  asepo asepi abe = hands and feet.

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

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

Illustrations of this kind might be multiplied almost indefinitely; and it
is well to note that they may be drawn from all parts of the world. South
America is peculiarly rich in native numeral words of this kind; and, as
the examples above cited show, it is the field to which one instinctively
turns when this subject is under discussion. The Zamuco numerals are, among
others, exceedingly interesting, giving us still a new variation in method.
They are[77]

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

As is here indicated, the form of progression from 5 to 10, which we should
expect to be "hand-1," or "hand-and-1," or some kindred expression,
signifying that one hand had been completed, is simply "1 on the other."
Again, the expressions for 11, 12, etc., are merely "1 on the foot," "2 on
the foot," etc., while 20 is "both feet ended."

An equally interesting scale is furnished by the language of the
Maipures[78] of the Orinoco, who count

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

In all the examples thus far given, 20 is expressed either by the
equivalent of "man" or by some formula introducing the word "feet." Both
these modes of expressing what our own ancestors termed a "score," are so
common that one hesitates to say which is of the more frequent use. The
following scale, from one of the Betoya dialects[79] of South America, is
quite remarkable among digital scales, making no use of either "man" or
"foot," but reckoning solely by fives, or hands, as the numerals indicate.

   1.  tey.
   2.  cayapa.
   3.  toazumba.
   4.  cajezea    = 2 with plural termination.
   5.  teente     = hand.
   6.  teyentetey = hand + 1.
   7.  teyente cayapa    = hand + 2.
   8.  teyente toazumba  = hand + 3.
   9.  teyente caesea    = hand + 4.
  10.  caya ente, or caya huena = 2 hands.
  11.  caya ente-tey     = 2 hands + 1.
  15.  toazumba-ente     = 3 hands.
  16.  toazumba-ente-tey = 3 hands + 1.
  20.  caesea ente       = 4 hands.

In the last chapter mention was made of the scanty numeral systems of the
Australian tribes, but a single scale was alluded to as reaching the
comparatively high limit of 20. This system is that belonging to the
Pikumbuls,[80] and the count runs thus:

   1.  mal.
   2.  bular.
   3.  guliba.
   4.  bularbular = 2-2.
   5.  mulanbu.
   6.  malmulanbu mummi        = 1 and 5 added on.
   7.  bularmulanbu mummi      = 2 and 5 added on.
   8.  gulibamulanbu mummi     = 3 and 5 added on.
   9.  bularbularmulanbu mummi = 4 and 5 added on.
  10.  bularin murra      = belonging to the 2 hands.
  11.  maldinna mummi     = 1 of the toes added on (to the 10 fingers).
  12.  bular dinna mummi  = 2 of the toes added on.
  13.  guliba dinna mummi = 3 of the toes added on.
  14.  bular bular dinna mummi = 4 of the toes added on.
  15.  mulanba dinna = 5 of the toes added on.
  16.  mal dinna mulanbu    = 1 and 5 toes.
  17.  bular dinna mulanbu  = 2 and 5 toes.
  18.  guliba dinna mulanbu = 3 and 5 toes.
  19.  bular bular dinna mulanbu = 4 and 5 toes.
  20.  bularin dinna = belonging to the 2 feet.

As has already been stated, there is good ground for believing that this
system was originally as limited as those obtained from other Australian
tribes, and that its extension from 4, or perhaps from 5 onward, is of
comparatively recent date.

A somewhat peculiar numeral nomenclature is found in the language of the
Klamath Indians of Oregon. The first ten words in the Klamath scale
are:[81]

   1.  nash, or nas.
   2.  lap   = hand.
   3.  ndan.
   4.  vunep = hand up.
   5.  tunep = hand away.
   6.  nadshkshapta = 1 I have bent over.
   7.  lapkshapta   = 2 I have bent over.
   8.  ndankshapta  = 3 I have bent over.
   9.  nadshskeksh  = 1 left over.
  10.  taunep       = hand hand?

In describing this system Mr. Gatschet says: "If the origin of the Klamath
numerals is thus correctly traced, their inventors must have counted only
the four long fingers without the thumb, and 5 was counted while saying
_hand away! hand off!_ The 'four,' or _hand high! hand up!_ intimates that
the hand was held up high after counting its four digits; and some term
expressing this gesture was, in the case of _nine_, substituted by 'one
left over' ... which means to say, 'only one is left until all the fingers
are counted.'" It will be observed that the Klamath introduces not only the
ordinary finger manipulation, but a gesture of the entire hand as well. It
is a common thing to find something of the kind to indicate the completion
of 5 or 10, and in one or two instances it has already been alluded to.
Sometimes one or both of the closed fists are held up; sometimes the open
hand, with all the fingers extended, is used; and sometimes an entirely
independent gesture is introduced. These are, in general, of no special
importance; but one custom in vogue among some of the prairie tribes of
Indians, to which my attention was called by Dr. J. Owen Dorsey,[82] should
be mentioned. It is a gesture which signifies multiplication, and is
performed by throwing the hand to the left. Thus, after counting 5, a wave
of the hand to the left means 50. As multiplication is rather unusual among
savage tribes, this is noteworthy, and would seem to indicate on the part
of the Indian a higher degree of intelligence than is ordinarily possessed
by uncivilized races.

In the numeral scale as we possess it in English, we find it necessary to
retain the name of the last unit of each kind used, in order to describe
definitely any numeral employed. Thus, fifteen, one hundred forty-two, six
thousand seven hundred twenty-seven, give in full detail the numbers they
are intended to describe. In primitive scales this is not always considered
necessary; thus, the Zamucos express their teens without using their word
for 10 at all. They say simply, 1 on the foot, 2 on the foot, etc.
Corresponding abbreviations are often met; so often, indeed, that no
further mention of them is needed. They mark one extreme, the extreme of
brevity, found in the savage method of building up hand, foot, and finger
names for numerals; while the Zuñi scale marks the extreme of prolixity in
the formation of such words. A somewhat ruder composition than any yet
noticed is shown in the numerals of the Vilelo scale,[83] which are:

   1.  agit, or yaagit.
   2.  uke.
   3.  nipetuei.
   4.  yepkatalet.
   5.  isig-nisle-yaagit    = hand fingers 1.
   6.  isig-teet-yaagit     = hand with 1.
   7.  isig-teet-uke        = hand with 2.
   8.  isig-teet-nipetuei   = hand with 3.
   9.  isig-teet-yepkatalet = hand with 4.
  10.  isig-uke-nisle = second hand fingers (lit. hand-two-fingers).
  11.  isig-uke-nisle-teet-yaagit = second hand fingers with 1.
  20.  isig-ape-nisle-lauel = hand foot fingers all.

In the examples thus far given, it will be noticed that the actual names of
individual fingers do not appear. In general, such words as thumb,
forefinger, little finger, are not found, but rather the hand-1, 1 on the
next, or 1 over and above, which we have already seen, are the type forms
for which we are to look. Individual finger names do occur, however, as in
the scale of the Hudson's Bay Eskimos,[84] where the three following words
are used both as numerals and as finger names:

   8.  kittukleemoot   = middle finger.
   9.  mikkeelukkamoot = fourth finger.
  10.  eerkitkoka      = little finger.

Words of similar origin are found in the original Jiviro scale,[85] where
the native numerals are:

   1.  ala.
   2.  catu.
   3.  cala.
   4.  encatu.
   5.  alacötegladu = 1 hand.
   6.  intimutu     = thumb (of second hand).
   7.  tannituna    = index finger.
   8.  tannituna cabiasu = the finger next the index finger.
   9.  bitin ötegla cabiasu = hand next to complete.
  10.  catögladu    = 2 hands.

As if to emphasize the rarity of this method of forming numerals, the
Jiviros afterward discarded the last five of the above scale, replacing
them by words borrowed from the Quichuas, or ancient Peruvians. The same
process may have been followed by other tribes, and in this way numerals
which were originally digital may have disappeared. But we have no evidence
that this has ever happened in any extensive manner. We are, rather,
impelled to accept the occasional numerals of this class as exceptions to
the general rule, until we have at our disposal further evidence of an
exact and critical nature, which would cause us to modify this opinion. An
elaborate philological study by Dr. J.H. Trumbull[86] of the numerals used
by many of the North American Indian tribes reveals the presence in the
languages of these tribes of a few, but only a few, finger names which are
used without change as numeral expressions also. Sometimes the finger gives
a name not its own to the numeral with which it is associated in
counting--as in the Chippeway dialect, which has _nawi-nindj_, middle of
the hand, and _nisswi_, 3; and the Cheyenne, where _notoyos_, middle
finger, and _na-nohhtu_, 8, are closely related. In other parts of the
world isolated examples of the transference of finger names to numerals are
also found. Of these a well-known example is furnished by the Zulu
numerals, where "_tatisitupa_, taking the thumb, becomes a numeral for six.
Then the verb _komba_, to point, indicating the forefinger, or 'pointer,'
makes the next numeral, seven. Thus, answering the question, 'How much did
your master give you?' a Zulu would say, '_U kombile_,' 'He pointed with
his forefinger,' _i.e._ 'He gave me seven'; and this curious way of using
the numeral verb is also shown in such an example as '_amahasi akombile_,'
'the horses have pointed,' _i.e._ 'there were seven of them.' In like
manner, _Kijangalobili_, 'keep back two fingers,' _i.e._ eight, and
_Kijangalolunje_, 'keep back one finger,' _i.e._ nine, lead on to _kumi_,
ten."[87]

Returning for a moment to the consideration of number systems in the
formation of which the influence of the hand has been paramount, we find
still further variations of the method already noticed of constructing
names for the fives, tens, and twenties, as well as for the intermediate
numbers. Instead of the simple words "hand," "foot," etc., we not
infrequently meet with some paraphrase for one or for all these terms, the
derivation of which is unmistakable. The Nengones,[88] an island tribe of
the Indian Ocean, though using the word "man" for 20, do not employ
explicit hand or foot words, but count

   1.  sa.
   2.  rewe.
   3.  tini.
   4.  etse.
   5.  se dono       = the end (of the first hand).
   6.  dono ne sa    = end and 1.
   7.  dono ne rewe  = end and 2.
   8.  dono ne tini  = end and 3.
   9.  dono ne etse  = end and 4.
  10.  rewe tubenine = 2 series (of fingers).
  11.  rewe tubenine ne sa re tsemene = 2 series and 1 on the next?
  20.  sa re nome    = 1 man.
  30.  sa re nome ne rewe tubenine = 1 man and 2 series.
  40.  rewe ne nome  = 2 men.

Examples like the above are not infrequent. The Aztecs used for 10 the word
_matlactli_, hand-half, _i.e._ the hand half of a man, and for 20
_cempoalli_, one counting.[89] The Point Barrow Eskimos call 10 _kodlin_,
the upper part, _i.e._ of a man. One of the Ewe dialects of Western
Africa[90] has _ewo_, done, for 10; while, curiously enough, 9, _asieke_,
is a digital word, meaning "to part (from) the hand."

In numerous instances also some characteristic word not of hand derivation
is found, like the Yoruba _ogodzi_, string, which becomes a numeral for 40,
because 40 cowries made a "string"; and the Maori _tekau_, bunch, which
signifies 10. The origin of this seems to have been the custom of counting
yams and fish by "bunches" of ten each.[91]

Another method of forming numeral words above 5 or 10 is found in the
presence of such expressions as second 1, second 2, etc. In languages of
rude construction and incomplete development the simple numeral scale is
often found to end with 5, and all succeeding numerals to be formed from
the first 5. The progression from that point may be 5-1, 5-2, etc., as in
the numerous quinary scales to be noticed later, or it may be second 1,
second 2, etc., as in the Niam Niam dialect of Central Africa, where the
scale is[92]

   1.  sa.
   2.  uwi.
   3.  biata.
   4.  biama.
   5.  biswi.
   6.  batissa = 2d 1.
   7.  batiwwi = 2d 2.
   8.  batti-biata = 2d 3.
   9.  batti-biama = 2d 4.
  10.  bauwé   = 2d 5.

That this method of progression is not confined to the least developed
languages, however, is shown by a most cursory examination of the numerals
of our American Indian tribes, where numeral formation like that exhibited
above is exceedingly common. In the Kootenay dialect,[93] of British
Columbia, _qaetsa_, 4, and _wo-qaetsa,_ 8, are obviously related, the
latter word probably meaning a second 4. Most of the native languages of
British Columbia form their words for 7 and 8 from those which signify 2
and 3; as, for example, the Heiltsuk,[94] which shows in the following
words a most obvious correspondence:

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

In the Choctaw language[95] the relation between 2 and 7, and 3 and 8, is
no less clear. Here the words are:

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

The Nez Percés[96] repeat the first three words of their scale in their 6,
7, and 8 respectively, as a comparison of these numerals will show.

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

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

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

A precisely similar method of numeral building is used by some of our
Western Indian tribes. Selecting a few of the Assiniboine numerals[98] as
an illustration, we have

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

A still more primitive structure is shown in the numerals of the
Mboushas[99] of Equatorial Africa. Instead of using 5-1, 5-2, 5-3, 5-4, or
2d 1, 2d 2, 2d 3, 2d 4, in forming their numerals from 6 to 9, they proceed
in the following remarkable and, at first thought, inexplicable manner to
form their compound numerals:

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

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


  BALAD.[100]                          UEA.[100]

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


Such examples are, I believe, entirely unique among primitive number
systems.

In numeral scales where the formative process has been of the general
nature just exhibited, irregularities of various kinds are of frequent
occurrence. Hand numerals may appear, and then suddenly disappear, just
where we should look for them with the greatest degree of certainty. In the
Ende,[101] a dialect of the Flores Islands, 5, 6, and 7 are of hand
formation, while 8 and 9 are of entirely different origin, as the scale
shows.

   1.  sa.
   2.  zua.
   3.  telu.
   4.  wutu.
   5.  lima
   6.  lima sa  = hand 1.
   7.  lima zua = hand 2.
   8.  rua butu = 2 × 4.
   9.  trasa    = 10 - 1?
  10.  sabulu.

One special point to be noticed in this scale is the irregularity that
prevails between 7, 8, 9. The formation of 7 is of the most ordinary kind;
8 is 2 fours--common enough duplication; while 9 appears to be 10 - 1. All
of these modes of compounding are, in their own way, regular; but the
irregularity consists in using all three of them in connective numerals in
the same system. But, odd as this jumble seems, it is more than matched by
that found in the scale of the Karankawa Indians,[102] an extinct tribe
formerly inhabiting the coast region of Texas. The first ten numerals of
this singular array are:

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

Systems like the above, where chaos instead of order seems to be the ruling
principle, are of occasional occurrence, but they are decidedly the
exception.

In some of the cases that have been adduced for illustration it is to be
noticed that the process of combination begins with 7 instead of with 6.
Among others, the scale of the Pigmies of Central Africa[103] and that of
the Mosquitos[104] of Central America show this tendency. In the Pigmy
scale the words for 1 and 6 are so closely akin that one cannot resist the
impression that 6 was to them a new 1, and was thus named.


       MOSQUITO.                           PIGMY.

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

The Mosquito scale is quite exceptional in forming 7, 8, and 9 from 6,
instead of from 5. The usual method, where combinations appear between 6
and 10, is exhibited by the Pigmy scale. Still another species of numeral
form, quite different from any that have already been noticed, is found in
the Yoruba[105] scale, which is in many respects one of the most peculiar
in existence. Here the words for 11, 12, etc., are formed by adding the
suffix _-la_, great, to the words for 1, 2, etc., thus:

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

The word for 40 was adopted because cowrie shells, which are used for
counting, were strung by forties; and _igba_, 200, because a heap of 200
shells was five strings, and thus formed a convenient higher unit for
reckoning. Proceeding in this curious manner,[106] they called 50 strings 1
_afo_ or head; and to illustrate their singular mode of reckoning--the king
of the Dahomans, having made war on the Yorubans, and attacked their army,
was repulsed and defeated with a loss of "two heads, twenty strings, and
twenty cowries" of men, or 4820.

The number scale of the Abipones,[107] one of the low tribes of the
Paraguay region, contains two genuine curiosities, and by reason of those
it deserves a place among any collection of numeral scales designed to
exhibit the formation of this class of words. It is:

   1.  initara = 1 alone.
   2.  inoaka.
   3.  inoaka yekaini = 2 and 1.
   4.  geyenknate     = toes of an ostrich.
   5.  neenhalek      = a five coloured, spotted hide,
        or hanambegen = fingers of 1 hand.
  10.  lanamrihegem   = fingers of both hands.
  20.  lanamrihegem cat gracherhaka anamichirihegem = fingers of both
       hands together with toes of both feet.

That the number sense of the Abipones is but little, if at all, above that
of the native Australian tribes, is shown by their expressing 3 by the
combination 2 and 1. This limitation, as we have already seen, is shared by
the Botocudos, the Chiquitos, and many of the other native races of South
America. But the Abipones, in seeking for words with which to enable
themselves to pass beyond the limit 3, invented the singular terms just
given for 4 and 5. The ostrich, having three toes in front and one behind
on each foot presented them with a living example of 3 + 1; hence "toes of
an ostrich" became their numeral for 4. Similarly, the number of colours in
a certain hide being five, the name for that hide was adopted as their next
numeral. At this point they began to resort to digital numeration also; and
any higher number is expressed by that method.

In the sense in which the word is defined by mathematicians, _number_ is a
pure, abstract concept. But a moment's reflection will show that, as it
originates among savage races, number is, and from the limitations of their
intellect must be, entirely concrete. An abstract conception is something
quite foreign to the essentially primitive mind, as missionaries and
explorers have found to their chagrin. The savage can form no mental
concept of what civilized man means by such a word as "soul"; nor would his
idea of the abstract number 5 be much clearer. When he says _five_, he
uses, in many cases at least, the same word that serves him when he wishes
to say _hand_; and his mental concept when he says _five_ is of a hand. The
concrete idea of a closed fist or an open hand with outstretched fingers,
is what is upper-most in his mind. He knows no more and cares no more about
the pure number 5 than he does about the law of the conservation of energy.
He sees in his mental picture only the real, material image, and his only
comprehension of the number is, "these objects are as many as the fingers
on my hand." Then, in the lapse of the long interval of centuries which
intervene between lowest barbarism and highest civilization, the abstract
and the concrete become slowly dissociated, the one from the other. First
the actual hand picture fades away, and the number is recognized without
the original assistance furnished by the derivation of the word. But the
number is still for a long time a certain number _of objects_, and not an
independent concept. It is only when the savage ceases to be wholly an
animal, and becomes a thinking human being, that number in the abstract can
come within the grasp of his mind. It is at this point that mere reckoning
ceases, and arithmetic begins.





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 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 than one may readily imagine that these numbers really were
"this" and "that" in the savage's original conception of them; and the same
likeness also occurs in the words for 3 and 4, which may readily enough
have been a second "this" and a second "that." In the Lushu tongue the
words for 1 and 2 are _tizi_ and _tazi_ respectively. In Koriak we find
_ngroka_, 3, and _ngraka_, 4; in Kolyma, _niyokh_, 3, and _niyakh_, 4; and
in Kamtschatkan, _tsuk_, 3, and _tsaak_, 4.[108] Sometimes, as in the case
of the Australian races, the entire extent of the count is carried through
by means of pairs. But the natural theory one would form is, that 2 is the
halting place for a very long time; that up to this point the fingers may
or may not have been used--probably not; and that when the next start is
made, and 3, 4, 5, and so on are counted, the fingers first come into
requisition. If the grammatical structure of the earlier languages of the
world's history is examined, the student is struck with the prevalence of
the dual number in them--something which tends to disappear as language
undergoes extended development. The dual number points unequivocally to the
time when 1 and 2 were _the_ numbers at mankind's disposal; to the time
when his three numeral concepts, 1, 2, many, each demanded distinct
expression. With increasing knowledge the necessity for this
differentiatuin would pass away, and but two numbers, singular and plural,
would remain. Incidentally it is to be noticed that the Indo-European words
for 3--_three_, _trois_, _drei_, _tres_, _tri,_ etc., have the same root as
the Latin _trans_, beyond, and give us a hint of the time when our Aryan
ancestors counted in the manner I have just described.

The first real difficulty which the savage experiences in counting, the
difficulty which comes when he attempts to pass beyond 2, and to count 3,
4, and 5, is of course but slight; and these numbers are commonly used and
readily understood by almost all tribes, no matter how deeply sunk in
barbarism we find them. But the instances that have already been cited must
not be forgotten. The Chiquitos do not, in their primitive state, properly
count at all; the Andamans, the Veddas, and many of the Australian tribes
have no numerals higher than 2; others of the Australians and many of the
South Americans stop with 3 or 4; and tribes which make 5 their limit are
still more numerous. Hence it is safe to assert that even this
insignificant number is not always reached with perfect ease. Beyond 5
primitive man often proceeds with the greatest difficulty. Most savages,
even those of the tribes just mentioned, can really count above here, even
though they have no words with which to express their thought. But they do
it with reluctance, and as they go on they quickly lose all sense of
accuracy. This has already been commented on, but to emphasize it afresh
the well-known example given by Mr. Oldfield from his own experience among
the Watchandies may be quoted.[109] "I once wished to ascertain the exact
number of natives who had been slain on a certain occasion. The individual
of whom I made the inquiry began to think over the names ... assigning one
of his fingers to each, and it was not until after many failures, and
consequent fresh starts, that he was able to express so high a number,
which he at length did by holding up his hand three times, thus giving me
to understand that fifteen was the answer to this most difficult
arithmetical question." This meagreness of knowledge in all things
pertaining to numbers is often found to be sharply emphasized in the names
adopted by savages for their numeral words. While discussing in a previous
chapter the limits of number systems, we found many instances where
anything above 2 or 3 was designated by some one of the comprehensive terms
_much_, _many_, _very many_; these words, or such equivalents as _lot_,
_heap_, or _plenty_, serving as an aid to the finger pantomime necessary to
indicate numbers for which they have no real names. The low degree of
intelligence and civilization revealed by such words is brought quite as
sharply into prominence by the word occasionally found for 5. Whenever the
fingers and hands are used at all, it would seem natural to expect for 5
some general expression signifying _hand_, for 10 _both hands_, and for 20
_man_. Such is, as we have already seen, the ordinary method of
progression, but it is not universal. A drop in the scale of civilization
takes us to a point where 10, instead of 20, becomes the whole man. The
Kusaies,[110] of Strong's Island, call 10 _sie-nul_, 1 man, 30 _tol-nul_, 3
men, 40 _a naul_, 4 men, etc.; and the Ku-Mbutti[111] of central Africa
have _mukko_, 10, and _moku_, man. If 10 is to be expressed by reference to
the man, instead of his hands, it might appear more natural to employ some
such expression as that adopted by the African Pigmies,[112] who call 10
_mabo_, and man _mabo-mabo_. With them, then, 10 is perhaps "half a man,"
as it actually is among the Towkas of South America; and we have already
seen that with the Aztecs it was _matlactli_, the "hand half" of a
man.[113] The same idea crops out in the expression used by the Nicobar
Islanders for 30--_heam-umdjome ruktei_, 1 man (and a) half.[114] Such
nomenclature is entirely natural, and it accords with the analogy offered
by other words of frequent occurrence in the numeral scales of savage
races. Still, to find 10 expressed by the term _man_ always conveys an
impression of mental poverty; though it may, of course, be urged that this
might arise from the fact that some races never use the toes in counting,
but go over the fingers again, or perhaps bring into requisition the
fingers of a second man to express the second 10. It is not safe to
postulate an extremely low degree of civilization from the presence of
certain peculiarities of numeral formation. Only the most general
statements can be ventured on, and these are always subject to modification
through some circumstance connected with environment, mode of living, or
intercourse with other tribes. Two South American races may be cited, which
seem in this respect to give unmistakable evidence of being sunk in deepest
barbarism. These are the Juri and the Cayriri, who use the same word for
man and for 5. The former express 5 by _ghomen apa_, 1 man,[115] and the
latter by _ibicho_, person.[116] The Tasmanians of Oyster Bay use the
native word of similar meaning, _puggana_, man,[117] for 5.

Wherever the numeral 20 is expressed by the term _man_, it may be expected
that 40 will be 2 men, 60, 3 men, etc. This form of numeration is usually,
though not always, carried as far as the system extends; and it sometimes
leads to curious terms, of which a single illustration will suffice. The
San Blas Indians, like almost all the other Central and South American
tribes, count by digit numerals, and form their twenties as follows:[118]

    20.  tula guena  = man 1.
    40.  tula pogua  = man 2.
   100.  tula atala  = man 5.
   120.  tula nergua = man 6.
  1000.  tula wala guena = great 1 man.

The last expression may, perhaps, be translated "great hundred," though the
literal meaning is the one given. If 10, instead of 20, is expressed by the
word "man," the multiples of 10 follow the law just given for multiples of
20. This is sufficiently indicated by the Kusaie scale; or equally well by
the Api words for 100 and 200, which are[119]

     _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 Greenlanders[120] 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 Mabas[121] call 10 _atuk_, great 1; the
Hottentots[122] and the Hidatsa Indians call 100 great 10, their words
being _gei disi_ and _pitikitstia_ respectively.

The Nicaraguans[123] express 100 by _guhamba_, great 10, and 400 by
_dinoamba_, great 20; and our own familiar word "million," which so many
modern languages have borrowed from the Italian, is nothing more nor less
than a derivative of the Latin _mille_, and really means "great thousand."
The Dakota[124] language shows the same origin for its expression of
1,000,000, which is _kick ta opong wa tunkah_, great 1000. The origin of
such terms can hardly be ascribed to poverty of language. It is found,
rather, in the mental association of the larger with the smaller unit, and
the consequent repetition of the name of the smaller. Any unit, whether it
be a single thing, a dozen, a score, a hundred, a thousand, or any other
unit, is, whenever used, a single and complete group; and where the
relation between them is sufficiently close, as in our "gross" and "great
gross," this form of nomenclature is natural enough to render it a matter
of some surprise that it has not been employed more frequently. An old
English nursery rhyme makes use of this association, only in a manner
precisely the reverse of that which appears now and then in numeral terms.
In the latter case the process is always one of enlargement, and the
associative word is "great." In the following rhyme, constructed by the
mature for the amusement of the childish mind, the process is one of
diminution, and the associative word is "little":

   One's none,
   Two's some,
   Three's a many,
   Four's a penny,
   Five's a little hundred.[125]

Any real numeral formation by the use of "little," with the name of some
higher unit, would, of course, be impossible. The numeral scale must be
complete before the nursery rhyme can be manufactured.

It is not to be supposed from the observations that have been made on the
formation of savage numeral scales that all, or even the majority of
tribes, proceed in the awkward and faltering manner indicated by many of
the examples quoted. Some of the North American Indian tribes have numeral
scales which are, as far as they go, as regular and almost as simple as our
own. But where digital numeration is extensively resorted to, the
expressions for higher numbers are likely to become complex, and to act as
a real bar to the extension of the system. The same thing is true, to an
even greater degree, of tribes whose number sense is so defective that they
begin almost from the outset to use combinations. If a savage expresses the
number 3 by the combination 2-1, it will at once be suspected that his
numerals will, by the time he reaches 10 or 20, become so complex and
confused that numbers as high as these will be expressed by finger
pantomime rather than by words. Such is often the case; and the comment is
frequently made by explorers that the tribes they have visited have no
words for numbers higher than 3, 4, 5, 10, or 20, but that counting is
carried beyond that point by the aid of fingers or other objects. So
reluctant, in many cases, are savages to count by words, that limits have
been assigned for spoken numerals, which subsequent investigation proved to
fall far short of the real extent of the number systems to which they
belonged. One of the south-western Indian tribes of the United States, the
Comanches, was for a time supposed to have no numeral words below 10, but
to count solely by the use of fingers. But the entire scale of this
taciturn tribe was afterward discovered and published.

To illustrate the awkward and inconvenient forms of expression which
abound in primitive numeral nomenclature, one has only to draw from such
scales as those of the Zuñi, or the Point Barrow Eskimos, given in the
last chapter.  Terms such as are found there may readily be duplicated
from almost any quarter of the globe. The Soussous of Sierra Leone[126]
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
is[127] _roambistelo polo amby valonjato amby arivo_, 2 + 30 + 800 + 1000.
The Aztec equivalent for 399 is[128] _caxtolli onnauh poalli ipan caxtolli
onnaui_, (15 + 4)  × 20 + 15 + 4; and the Sioux require for 29 the
ponderous combination[129] _wick a chimen ne nompah sam pah nep e chu wink
a._ These terms, long and awkward as they seem, are only the legitimate
results which arise from combining the names of the higher and lower
numbers, according to the peculiar genius of each language. From some of
the Australian tribes are derived expressions still more complex, as for
6, _marh-jin-bang-ga-gudjir-gyn_, half the hands and 1; and for 15,
_marh-jin-belli-belli-gudjir-jina-bang-ga_, the hand on either side and
half the feet.[130] The Maré tribe, one of the numerous island tribes of
Melanesia,[131] required for a translation of the numeral 38, which occurs
in John v. 5, "had an infirmity thirty and eight years," the
circumlocution, "one man and both sides five and three." Such expressions,
curious as they seem at first thought, are no more than the natural
outgrowth of systems built up by the slow and tedious process which so
often obtains among primitive races, where digit numerals are combined in
an almost endless variety of ways, and where mere reduplication often
serves in place of any independent names for higher units. To what extent
this may be carried is shown by the language of the Cayubabi,[132] who have
for 10 the word _tunca_, and for 100 and 1000 the compounds _tunca tunca_,
and _tunca tunca tunca_ respectively; or of the Sapibocones, who call 10
_bururuche_, hand hand, and 100 _buruche buruche_, hand hand hand
hand.[133] More remarkable still is the Ojibwa language, which continues
its numeral scale without limit, furnishing combinations which are really
remarkable; as, _e.g._, that for 1,000,000,000, which is _me das wac me das
wac as he me das wac_,[134] 1000 × 1000  × 1000. The Winnebago expression
for the same number,[135] _ho ke he hhuta hhu chen a ho ke he ka ra pa ne
za_ is no less formidable, but it has every appearance of being an honest,
native combination. All such primitive terms for larger numbers must,
however, be received with caution. Savages are sometimes eager to display a
knowledge they do not possess, and have been known to invent numeral words
on the spot for the sake of carrying their scales to as high a limit as
possible. The Choctaw words for million and billion are obvious attempts to
incorporate the corresponding English terms into their own language.[136]
For million they gave the vocabulary-hunter the phrase _mil yan chuffa_,
and for billion, _bil yan chuffa_. The word _chuffa_ signifies 1, hence
these expressions are seen at a glance to be coined solely for the purpose
of gratifying a little harmless Choctaw vanity. But this is innocence
itself compared with the fraud perpetrated on Labillardière by the Tonga
Islanders, who supplied the astonished and delighted investigator with a
numeral vocabulary up to quadrillions. Their real limit was afterward found
to be 100,000, and above that point they had palmed off as numerals a
tolerably complete list of the obscene words of their language, together
with a few nonsense terms. These were all accepted and printed in good
faith, and the humiliating truth was not discovered until years
afterward.[137]

One noteworthy and interesting fact relating to numeral nomenclature is the
variation in form which words of this class undergo when applied to
different classes of objects. To one accustomed as we are to absolute and
unvarying forms for numerals, this seems at first a novel and almost
unaccountable linguistic freak. But it is not uncommon among uncivilized
races, and is extensively employed by so highly enlightened a people, even,
as the Japanese. This variation in form is in no way analogous to that
produced by inflectional changes, such as occur in Hebrew, Greek, Latin,
etc. It is sufficient in many cases to produce almost an entire change in
the form of the word; or to result in compounds which require close
scrutiny for the detection of the original root. For example, in the
Carrier, one of the Déné dialects of western Canada, the word _tha_ means 3
things; _thane_, 3 persons; _that_, 3 times; _thatoen_, in 3 places;
_thauh_, in 3 ways; _thailtoh_, all of the 3 things; _thahoeltoh_, all of
the 3 persons; and _thahultoh_, all of the 3 times.[138] In the Tsimshian
language of British Columbia we find seven distinct sets of numerals "which
are used for various classes of objects that are counted. The first set is
used in counting where there is no definite object referred to; the second
class is used for counting flat objects and animals; the third for counting
round objects and divisions of time; the fourth for counting men; the fifth
for counting long objects, the numerals being composed with _kan_, tree;
the sixth for counting canoes; and the seventh for measures. The last seem
to be composed with _anon_, hand."[139] The first ten numerals of each of
these classes is given in the following table:

 +----+---------+---------+---------+----------+------------+-------------+-------------+
 |No. |Counting | Flat    | 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'alp[=e]eltk|gulbel       |
 | 3  |guant    |guant    |gutle    |gulal     |galtskan    |galtskantk   |guleont      |
 | 4  |tqalpq   |tqalpq   |tqalpq   |tqalpqdal |tqaapskan   |tqalpqsk     |tqalpqalont  |
 | 5  |kct[=o]nc|kct[=o]nc|kct[=o]nc|kcenecal  |k'etoentskan|kct[=o]onsk  |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    |kp[=e]el |kpal      |kp[=e]etskan|gy'apsk      |kpeont       |
 +----+---------+---------+---------+----------+------------+-------------+-------------+

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

 +-----------------------+-------------+--------------+--------------+
 |                       | One.        | Two.         | Three.       |
 +-----------------------+-------------+--------------+--------------+
 |Animate.               |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   |
 |Together on beach.     |----         |maalis        |----          |
 |Together in house, etc.|----         |maalitl       |----          |
 +-----------------------+-------------+--------------+--------------+

Variation in numeral forms such as is exhibited in the above tables is not
confined to any one quarter of the globe; but it is more universal among
the British Columbian Indians than among any other race, and it is a more
characteristic linguistic peculiarity of this than of any other region,
either in the Old World or in the New. It was to some extent employed by
the Aztecs,[142] and its use is current among the Japanese; in whose
language Crawfurd finds fourteen different classes of numerals "without
exhausting the list."[143]

In examining the numerals of different languages it will be found that the
tens of any ordinary decimal scale are formed in the same manner as in
English. Twenty is simply 2 times 10; 30 is 3 times 10, and so on. The word
"times" is, of course, not expressed, any more than in English; but the
expressions briefly are, 2 tens, 3 tens, etc. But a singular exception to
this method is presented by the Hebrew, and other of the Semitic languages.
In Hebrew the word for 20 is the plural of the word for 10; and 30, 40, 50,
etc. to 90 are plurals of 3, 4, 5, 6, 7, 8, 9. These numerals are as
follows:[144]

  10,  eser,     20,  eserim,
   3,  shalosh,  30,  shaloshim,
   4,  arba,     40,  arbaim,
   5,  chamesh,  50,  chamishshim,
   6,  shesh,    60,  sheshshim,
   7,  sheba,    70,  shibim,
   8,  shemoneh  80,  shemonim,
   9,  tesha,    90,  tishim.

The same formation appears in the numerals of the ancient Phoenicians,[145]
and seems, indeed, to be a well-marked characteristic of the various
branches of this division of the Caucasian race. An analogous method
appears in the formation of the tens in the Bisayan,[146] one of the Malay
numeral scales, where 30, 40, ... 90, are constructed from 3, 4, ... 9, by
adding the termination _-an_.

No more interesting contribution has ever been made to the literature of
numeral nomenclature than that in which Dr. Trumbull embodies the results
of his scholarly research among the languages of the native Indian tribes
of this country.[147] As might be expected, we are everywhere confronted
with a digital origin, direct or indirect, in the great body of the words
examined. But it is clearly shown that such a derivation cannot be
established for all numerals; and evidence collected by the most recent
research fully substantiates the position taken by Dr. Trumbull. Nearly all
the derivations established are such as to remind us of the meanings we
have already seen recurring in one form or another in language after
language. Five is the end of the finger count on one hand--as, the Micmac
_nan_, and Mohegan _nunon_, gone, or spent; the Pawnee _sihuks_, hands
half; the Dakota _zaptan_, hand turned down; and the Massachusetts
_napanna_, on one side. Ten is the end of the finger count, but is not
always expressed by the "both hands" formula so commonly met with. The Cree
term for this number is _mitatat_, no further; and the corresponding word
in Delaware is _m'tellen_, no more. The Dakota 10 is, like its 5, a
straightening out of the fingers which have been turned over in counting,
or _wickchemna_, spread out unbent. The same is true of the Hidatsa
_pitika_, which signifies a smoothing out, or straightening. The Pawnee 4,
_skitiks_, is unusual, signifying as it does "all the fingers," or more
properly, "the fingers of the hand." The same meaning attaches to this
numeral in a few other languages also, and reminds one of the habit some
people have of beginning to count on the forefinger and proceeding from
there to the little finger. Can this have been the habit of the tribes in
question? A suggestion of the same nature is made by the Illinois and Miami
words for 8, _parare_ and _polane_, which signify "nearly ended." Six is
almost always digital in origin, though the derivation may be indirect, as
in the Illinois _kakatchui_, passing beyond the middle; and the Dakota
_shakpe_, 1 in addition. Some of these significations are well matched by
numerals from the Ewe scales of western Africa, where we find the
following:[148]

   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 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
respectively[153]--words as natural and as much to be expected at the
beginning of a number scale as any finger name could possibly be. Among
almost all savage races one form or another of palpable arithmetic is
found, such as counting by seeds, pebbles, shells, notches, or knots; and
the derivation of number words from these sources can constitute no ground
for surprise. The Marquesan word for 4 is _pona_, knot, from the practice
of tying breadfruit in knots of 4. The Maori 10 is _tekau_, bunch, or
parcel, from the counting of yams and fish by parcels of 10.[154] The
Javanese call 25, _lawe_, a thread, or string; 50, _ekat_, a skein of
thread; 400, _samas_, a bit of gold; 800, _domas_, 2 bits of gold.[155] The
Macassar and Butong term for 100 is _bilangan_, 1 tale or reckoning.[156]
The Aztec 20 is _cem pohualli_, 1 count; 400 is _centzontli_, 1 hair of the
head; and 8000 is _xiquipilli_, sack.[157] This sack was of such a size as
to contain 8000 cacao nibs, or grains, hence the derivation of the word in
its numeral sense is perfectly natural. In Japanese we find a large number
of terms which, as applied to the different units of the number scale, seem
almost purely fanciful. These words, with their meanings as given by a
Japanese lexicon, are as follows:

  10,000, or 10^4,  män   = enormous number.
             10^8,  oku   = a compound of the words "man" and "mind."
             10^12, chio  = indication, or symptom.
             10^16, kei   = capital city.
             10^20, si    = a term referring to grains.
             10^24, owi   = ----
             10^28, jio   = extent of land.
             10^32, ko    = canal.
             10^36, kan   = some kind of a body of water.
             10^40, sai   = justice.
             10^44, s[=a] = support.
             10^48, kioku = limit, or more strictly, ultimate.
            .01^2,  rin   = ----
            .01^3,  mo    = hair (of some animal).
            .01^4,  shi   = thread.

In addition to these, some of the lower fractional values are described by
words meaning "very small," "very fine thread," "sand grain," "dust," and
"very vague." Taken altogether, the Japanese number system is the most
remarkable I have ever examined, in the extent and variety of the higher
numerals with well-defined descriptive names. Most of the terms employed
are such as to defy any attempt to trace the process of reasoning which led
to their adoption. It is not improbable that the choice was, in some of
these cases at least, either accidental or arbitrary; but still, the
changes in word meanings which occur with the lapse of time may have
differentiated significations originally alike, until no trace of kinship
would appear to the casual observer. Our numerals "score" and "gross" are
never thought of as having any original relation to what is conveyed by the
other meanings which attach to these words. But the origin of each, which
is easily traced, shows that, in the beginning, there existed a
well-defined reason for the selection of these, rather than other terms,
for the numbers they now describe. Possibly these remarkable Japanese terms
may be accounted for in the same way, though the supposition is, for some
reasons, quite improbable. The same may be said for the Malagasy 1000,
_alina_, which also means "night," and the Hebrew 6, _shesh_, which has the
additional signification "white marble," and the stray exceptions which now
and then come to the light in this or that language. Such terms as these
may admit of some logical explanation, but for the great mass of numerals
whose primitive meanings can be traced at all, no explanation whatever is
needed; the words are self-explanatory, as the examples already cited show.

A few additional examples of natural derivation may still further emphasize
the point just discussed. In Bambarese the word for 10, _tank_, is derived
directly from _adang_, to count.[158] In the language of Mota, one of the
islands of Melanesia, 100 is _mel nol_, used and done with, referring to
the leaves of the cycas tree, with which the count had been carried
on.[159] In many other Melanesian dialects[160] 100 is _rau_, a branch or
leaf. In the Torres Straits we find the same number expressed by _na won_,
the close; and in Eromanga it is _narolim narolim_ (2 × 5)(2 × 5).[161]
This combination deserves remark only because of the involved form which
seems to have been required for the expression of so small a number as 100.
A compound instead of a simple term for any higher unit is never to be
wondered at, so rude are some of the savage methods of expressing number;
but "two fives (times) two fives" is certainly remarkable. Some form like
that employed by the Nusqually[162] of Puget Sound for 1000, i.e.
_paduts-subquätche_, ten hundred, is more in accordance with primitive
method. But we are equally likely to find such descriptive phrases for this
numeral as the _dor paka_, banyan roots, of the Torres Islands; _rau na
hai_, leaves of a tree, of Vaturana; or _udolu_, all, of the Fiji Islands.
And two curious phrases for 1000 are those of the Banks' Islands, _tar
mataqelaqela_, eye blind thousand, _i.e._ many beyond count; and of
Malanta, _warehune huto_, opossum's hairs, or _idumie one_, count the
sand.[163]

The native languages of India, Thibet, and portions of the Indian
archipelago furnish us with abundant instances of the formation of
secondary numeral scales, which were used only for special purposes, and
without in any way interfering with the use of the number words already in
use. "Thus the scholars of India, ages ago, selected a set of words for a
memoria technica, in order to record dates and numbers. These words they
chose for reasons which are still in great measure evident; thus 'moon' or
'earth' expressed 1, there being but one of each; 2 might be called 'eye,'
'wing,' 'arm,' 'jaw,' as going in pairs; for 3 they said 'Rama,' 'fire,' or
'quality,' there being considered to be three Ramas, three kinds of fire,
three qualities (guna); for 4 were used 'veda,' 'age,' or 'ocean,' there
being four of each recognized; 'season' for 6, because they reckoned six
seasons; 'sage' or 'vowel,' for 7, from the seven sages and the seven
vowels; and so on with higher numbers, 'sun' for 12, because of his twelve
annual denominations, or 'zodiac' from his twelve signs, and 'nail' for 20,
a word incidentally bringing in finger notation. As Sanskrit is very rich
in synonyms, and as even the numerals themselves might be used, it became
very easy to draw up phrases or nonsense verses to record series of numbers
by this system of artificial memory."[164]

More than enough has been said to show how baseless is the claim that all
numeral words are derived, either directly or indirectly, from the names of
fingers, hands, or feet. Connected with the origin of each number word
there may be some metaphor, which cannot always be distinctly traced; and
where the metaphor was born of the hand or of the foot, we inevitably
associate it with the practice of finger counting. But races as fond of
metaphor and of linguistic embellishment as are those of the East, or as
are our American Indians even, might readily resort to some other source
than that furnished by the members of the human body, when in want of a
term with which to describe the 5, 10, or any other number of the numeral
scale they were unconsciously forming. That the first numbers of a numeral
scale are usually derived from other sources, we have some reason to
believe; but that all above 2, 3, or at most 4, are almost universally of
digital origin we must admit. Exception should properly be made of higher
units, say 1000 or anything greater, which could not be expected to conform
to any law of derivation governing the first few units of a system.

Collecting together and comparing with one another the great mass of terms
by which we find any number expressed in different languages, and, while
admitting the great diversity of method practised by different tribes, we
observe certain resemblances which were not at first supposed to exist. The
various meanings of 1, where they can be traced at all, cluster into a
little group of significations with which at last we come to associate the
idea of unity. Similarly of 2, or 5, or 10, or any one of the little band
which does picket duty for the advance guard of the great host of number
words which are to follow. A careful examination of the first decade
warrants the assertion that the probable meaning of any one of the units
will be found in the list given below. The words selected are intended
merely to serve as indications of the thought underlying the savage's
choice, and not necessarily as the exact term by means of which he
describes his number. Only the commonest meanings are included in the
tabulation here given.

   1   = existence, piece, group, beginning.
   2   = repetition, division, natural pair.
   3   = collection, many, two-one.
   4   = two twos.
   5   = hand, group, division,
   6   = five-one, two threes, second one.
   7   = five-two, second two, three from ten.
   8   = five-three, second three, two fours, two from ten.
   9   = five-four, three threes, one from ten.
  10   = one (group), two fives (hands), half a man, one man.
  15   = ten-five, one foot, three fives.
  20   = two tens, one man, two feet.[165]





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 ten-one. When this step has been taken, the base is established. The
savage has, with entire unconsciousness, made all his subsequent progress
dependent on the number 10, or, in other words, he has established 10 as
the base of his number system. The process just indicated may be gone
through with at 5, or at 20, thus giving us a quinary or a vigesimal, or,
more probably, a mixed system; and, in rare instances, some other number
may serve as the point of departure from simple into compound numeral
terms. But the general idea is always the same, and only the details of
formation are found to differ.

Without the establishment of some base any _system_ of numbers is
impossible. The savage has no means of keeping track of his count unless he
can at each step refer himself to some well-defined milestone in his
course. If, as has been pointed out in the foregoing chapters, confusion
results whenever an attempt is made to count any number which carries him
above 10, it must at once appear that progress beyond that point would be
rendered many times more difficult if it were not for the fact that, at
each new step, he has only to indicate the distance he has progressed
beyond his base, and not the distance from his original starting-point.
Some idea may, perhaps, be gained of the nature of this difficulty by
imagining the numbers of our ordinary scale to be represented, each one by
a single symbol different from that used to denote any other number. How
long would it take the average intellect to master the first 50 even, so
that each number could without hesitation be indicated by its appropriate
symbol? After the first 50 were once mastered, what of the next 50? and the
next? and the next? and so on. The acquisition of a scale for which we had
no other means of expression than that just described would be a matter of
the extremest difficulty, and could never, save in the most exceptional
circumstances, progress beyond the attainment of a limit of a few hundred.
If the various numbers in question were designated by words instead of by
symbols, the difficulty of the task would be still further increased.
Hence, the establishment of some number as a base is not only a matter of
the very highest convenience, but of absolute necessity, if any save the
first few numbers are ever to be used.

In the selection of a base,--of a number from which he makes a fresh start,
and to which he refers the next steps in his count,--the savage simply
follows nature when he chooses 10, or perhaps 5 or 20. But it is a matter
of the greatest interest to find that other numbers have, in exceptional
cases, been used for this purpose. Two centuries ago the distinguished
philosopher and mathematician, Leibnitz, proposed a binary system of
numeration. The only symbols needed in such a system would be 0 and 1. The
number which is now symbolized by the figure 2 would be represented by 10;
while 3, 4, 5, 6, 7, 8, etc., would appear in the binary notation as 11,
100, 101, 110, 111, 1000, etc. The difficulty with such a system is that it
rapidly grows cumbersome, requiring the use of so many figures for
indicating any number. But Leibnitz found in the representation of all
numbers by means of the two digits 0 and 1 a fitting symbolization of the
creation out of chaos, or nothing, of the entire universe by the power of
the Deity. In commemoration of this invention a medal was struck bearing on
the obverse the words

     Numero Deus impari gaudet,

and on the reverse,

     Omnibus ex nihilo ducendis sufficit Unum.[166]

This curious system seems to have been regarded with the greatest affection
by its inventor, who used every endeavour in his power to bring it to the
notice of scholars and to urge its claims. But it appears to have been
received with entire indifference, and to have been regarded merely as a
mathematical curiosity.

Unknown to Leibnitz, however, a binary method of counting actually existed
during that age; and it is only at the present time that it is becoming
extinct. In Australia, the continent that is unique in its flora, its
fauna, and its general topography, we find also this anomaly among methods
of counting. The natives, who are to be classed among the lowest and the
least intelligent of the aboriginal races of the world, have number systems
of the most rudimentary nature, and evince a decided tendency to count by
twos. This peculiarity, which was to some extent shared by the Tasmanians,
the island tribes of the Torres Straits, and other aboriginal races of that
region, has by some writers been regarded as peculiar to their part of the
world; as though a binary number system were not to be found elsewhere.
This attempt to make out of the rude and unusual method of counting which
obtained among the Australians a racial characteristic is hardly justified
by fuller investigation. Binary number systems, which are given in full on
another page, are found in South America. Some of the Dravidian scales are
binary;[167] and the marked preference, not infrequently observed among
savage races, for counting by pairs, is in itself a sufficient refutation
of this theory. Still it is an unquestionable fact that this binary
tendency is more pronounced among the Australians than among any other
extensive number of kindred races. They seldom count in words above 4, and
almost never as high as 7. One of the most careful observers among them
expresses his doubt as to a native's ability to discover the loss of two
pins, if he were first shown seven pins in a row, and then two were removed
without his knowledge.[168] But he believes that if a single pin were
removed from the seven, the Blackfellow would become conscious of its loss.
This is due to his habit of counting by pairs, which enables him to
discover whether any number within reasonable limit is odd or even. Some of
the negro tribes of Africa, and of the Indian tribes of America, have the
same habit. Progression by pairs may seem to some tribes as natural as
progression by single units. It certainly is not at all rare; and in
Australia its influence on spoken number systems is most apparent.

Any number system which passes the limit 10 is reasonably sure to have
either a quinary, a decimal, or a vigesimal structure. A binary scale
could, as it is developed in primitive languages, hardly extend to 20, or
even to 10, without becoming exceedingly cumbersome. A binary scale
inevitably suggests a wretchedly low degree of mental development, which
stands in the way of the formation of any number scale worthy to be
dignified by the name of system. Take, for example, one of the dialects
found among the western tribes of the Torres Straits, where, in general,
but two numerals are found to exist. In this dialect the method of counting
is:[169]

   1.  urapun.
   2.  okosa.
   3.  okosa urapun = 2-1.
   4.  okosa okosa  = 2-2.
   5.  okosa okosa urapun = 2-2-1.
   6.  okosa okosa okosa  = 2-2-2.

Anything above 6 they call _ras_, a lot.

For the sake of uniformity we may speak of this as a "system." But in so
doing, we give to the legitimate meaning of the word a severe strain. The
customs and modes of life of these people are not such as to require the
use of any save the scanty list of numbers given above; and their mental
poverty prompts them to call 3, the first number above a single pair, 2-1.
In the same way, 4 and 6 are respectively 2 pairs and 3 pairs, while 5 is 1
more than 2 pairs. Five objects, however, they sometimes denote by
_urapuni-getal_, 1 hand. A precisely similar condition is found to prevail
respecting the arithmetic of all the Australian tribes. In some cases only
two numerals are found, and in others three. But in a very great number of
the native languages of that continent the count proceeds by pairs, if
indeed it proceeds at all. Hence we at once reject the theory that
Australian arithmetic, or Australian counting, is essentially peculiar. It
is simply a legitimate result, such as might be looked for in any part of
the world, of the barbarism in which the races of that quarter of the world
were sunk, and in which they were content to live.

The following examples of Australian and Tasmanian number systems show how
scanty was the numerical ability possessed by these tribes, and illustrate
fully their tendency to count by twos or pairs.


  MURRAY RIVER.[170]

  1.   enea.
  2.   petcheval.
  3.   petchevalenea       = 2-1.
  4.   petcheval peteheval = 2-2.


  MAROURA.

  1.   nukee.
  2.   barkolo.
  3.   barkolo nuke    = 2-1.
  4.   barkolo barkolo = 2-2.


  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.


  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.


  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.


  GONN STATION.

  1.   karp.
  2.   pellige.
  3.   pellige karp    = 2-1.
  4.   pellige pellige = 2-2.


  UPPER YARRA.

  1.   kaambo.
  2.   benjero.
  3.   benjero kaambo     = 2-2.
  4.   benjero on benjero = 2-2.


  OMEO.

  1.   bore.
  2.   warkolala.
  3.   warkolala bore      = 2-1.
  4.   warkolala warkolala = 2-2.


  SNOWY RIVER.

  1.   kootook.
  2.   boolong.
  3.   booloom catha kootook = 2 + 1.
  4.   booloom catha booloom = 2 + 2.

  NGARRIMOWRO.
  1.   warrangen.
  2.   platir.
  3.   platir warrangen = 2-1.
  4.   platir platir    = 2-2.

This Australian list might be greatly extended, but the scales selected may
be taken as representative examples of Australian binary scales. Nearly all
of them show a structure too clearly marked to require comment. In a few
cases, however, the systems are to be regarded rather as showing a trace of
binary structure, than as perfect examples of counting by twos. Examples of
this nature are especially numerous in Curr's extensive list--the most
complete collection of Australian vocabularies ever made.

A few binary scales have been found in South America, but they show no
important variation on the Australian systems cited above. The only ones I
have been able to collect are the following:


  BAKAIRI.[188]

  1.   tokalole.
  2.   asage.
  3.   asage tokalo = 2-1.
  4.   asage asage  = 2-2.


  ZAPARA.[189]

  1.   nuquaqui.
  2.   namisciniqui.
  3.   haimuckumarachi.
  4.   namisciniqui ckara maitacka          = 2 + 2.
  5.   namisciniqui ckara maitacka nuquaqui = 2 pairs + 1.
  6.   haimuckumaracki ckaramsitacka        = 3 pairs.


  APINAGES.[190]

  1.   pouchi.
  2.   at croudou.
  3.   at croudi-pshi  = 2-1.
  4.   agontad-acroudo = 2-2.


  COTOXO.[191]

  1.   ihueto.
  2.   ize.
  3.   ize-te-hueto    = 2-1.
  4.   ize-te-seze     = 2-2.
  5.   ize-te-seze-hue = 2-2-1.


  MBAYI.[192]

  1.   uninitegui.
  2.   iniguata.
  3.   iniguata dugani     = 2 over.
  4.   iniguata driniguata = 2-2.
  5.   oguidi              = many.


  TAMA.[193]

  1.   teyo.
  2.   cayapa.
  3.   cho-teyo   = 2 + 1.
  4.   cayapa-ria = 2 again.
  5.   cia-jente  = hand.


  CURETU.[194]

  1.   tchudyu.
  2.   ap-adyu.
  3.   arayu.
  4.   apaedyái = 2 + 2.
  5.   tchumupa.

If the existence of number systems like the above are to be accounted for
simply on the ground of low civilization, one might reasonably expect to
find ternary and and quaternary scales, as well as binary. Such scales
actually exist, though not in such numbers as the binary. An example of the
former is the Betoya scale,[195] which runs thus:

  1.   edoyoyoi.
  2.   edoi           = another.
  3.   ibutu          = beyond.
  4.   ibutu-edoyoyoi = beyond 1, or 3-1.
  5.   ru-mocoso      = hand.

The Kamilaroi scale, given as an example of binary formation, is partly
ternary; and its word for 6, _guliba guliba_, 3-3, is purely ternary. An
occasional ternary trace is also found in number systems otherwise decimal
or quinary vigesimal; as the _dlkunoutl_, second 3, of the Haida Indians of
British Columbia. The Karens of India[196] in a system otherwise strictly
decimal, exhibit the following binary-ternary-quaternary vagary:

  6.   then tho    = 3 × 2.
  7.   then tho ta = 3 × 2-1.
  8.   lwie tho    = 4 × 2.
  9.   lwie tho ta = 4 × 2-1.

In the Wokka dialect,[197] found on the Burnett River, Australia, a single
ternary numeral is found, thus:

  1.   karboon.
  2.   wombura.
  3.   chrommunda.
  4.   chrommuda karboon = 3-1.

Instances of quaternary numeration are less rare than are those of ternary,
and there is reason to believe that this method of counting has been
practised more extensively than any other, except the binary and the three
natural methods, the quinary, the decimal, and the vigesimal. The number of
fingers on one hand is, excluding the thumb, four. Possibly there have been
tribes among which counting by fours arose as a legitimate, though unusual,
result of finger counting; just as there are, now and then, individuals who
count on their fingers with the forefinger as a starting-point. But no such
practice has ever been observed among savages, and such theorizing is the
merest guess-work. Still a definite tendency to count by fours is sometimes
met with, whatever be its origin. Quaternary traces are repeatedly to be
found among the Indian languages of British Columbia. In describing the
Columbians, Bancroft says: "Systems of numeration are simple, proceeding by
fours, fives, or tens, according to the different languages...."[198] The
same preference for four is said to have existed in primitive times in the
languages of Central Asia, and that this form of numeration, resulting in
scores of 16 and 64, was a development of finger counting.[199]

In the Hawaiian and a few other languages of the islands of the central
Pacific, where in general the number systems employed are decimal, we find
a most interesting case of the development, within number scales already
well established, of both binary and quaternary systems. Their origin seems
to have been perfectly natural, but the systems themselves must have been
perfected very slowly. In Tahitian, Rarotongan, Mangarevan, and other
dialects found in the neighbouring islands of those southern latitudes,
certain of the higher units, _tekau_, _rau_, _mano_, which originally
signified 10, 100, 1000, have become doubled in value, and now stand for
20, 200, 2000. In Hawaiian and other dialects they have again been doubled,
and there they stand for 40, 400, 4000.[200] In the Marquesas group both
forms are found, the former in the southern, the latter in the northern,
part of the archipelago; and it seems probable that one or both of these
methods of numeration are scattered somewhat widely throughout that region.
The origin of these methods is probably to be found in the fact that, after
the migration from the west toward the east, nearly all the objects the
natives would ever count in any great numbers were small,--as yams,
cocoanuts, fish, etc.,--and would be most conveniently counted by pairs.
Hence the native, as he counted one pair, two pairs, etc., might readily
say _one_, _two_, and so on, omitting the word "pair" altogether. Having
much more frequent occasion to employ this secondary than the primary
meaning of his numerals, the native would easily allow the original
significations to fall into disuse, and in the lapse of time to be entirely
forgotten. With a subsequent migration to the northward a second
duplication might take place, and so produce the singular effect of giving
to the same numeral word three different meanings in different parts of
Oceania. To illustrate the former or binary method of numeration, the
Tahuatan, one of the southern dialects of the Marquesas group, may be
employed.[201] Here the ordinary numerals are:

          1.  tahi,
         10.  onohuu.
         20.  takau.
        200.  au.
      2,000.  mano.
     20,000.  tini.
     20,000.  tufa.
  2,000,000.  pohi.

In counting fish, and all kinds of fruit, except breadfruit, the scale
begins with _tauna_, pair, and then, omitting _onohuu_, they employ the
same words again, but in a modified sense. _Takau_ becomes 10, _au_ 100,
etc.; but as the word "pair" is understood in each case, the value is the
same as before. The table formed on this basis would be:

   2 (units) = 1 tauna = 2.
  10 tauna   = 1 takau = 20.
  10 takau   = 1 au    = 200.
  10 au      = 1 mano  = 2000.
  10 mano    = 1 tini  = 20,000.
  10 tini    = 1 tufa  = 200,000.
  10 tufa    = 1 pohi  = 2,000,000.

For counting breadfruit they use _pona_, knot, as their unit, breadfruit
usually being tied up in knots of four. _Takau_ now takes its third
signification, 40, and becomes the base of their breadfruit system, so to
speak. For some unknown reason the next unit, 400, is expressed by _tauau_,
while _au_, which is the term that would regularly stand for that number,
has, by a second duplication, come to signify 800. The next unit, _mano_,
has in a similar manner been twisted out of its original sense, and in
counting breadfruit is made to serve for 8000. In the northern, or
Nukuhivan Islands, the decimal-quaternary system is more regular. It is in
the counting of breadfruit only,[202]

   4 breadfruits = 1 pona = 4.
  10 pona        = 1 toha = 40.
  10 toha        = 1 au   = 400.
  10 au          = 1 mano = 4000.
  10 mano        = 1 tini = 40,000.
  10 tini        = 1 tufa = 400,000.
  10 tufa        = 1 pohi = 4,000,000.

In the Hawaiian dialect this scale is, with slight modification, the
universal scale, used not only in counting breadfruit, but any other
objects as well. The result is a complete decimal-quaternary system, such
as is found nowhere else in the world except in this and a few of the
neighbouring dialects of the Pacific. This scale, which is almost identical
with the Nukuhivan, is[203]

   4 units  = 1 ha or tauna = 4.
  10 tauna  = 1 tanaha      = 40.
  10 tanaha = 1 lau         = 400.
  10 lau    = 1 mano        = 4000.
  10 mano   = 1 tini        = 40,000.
  10 tini   = 1 lehu        = 400,000.

The quaternary element thus introduced has modified the entire structure of
the Hawaiian number system. Fifty is _tanaha me ta umi_, 40 + 10; 76 is 40
+ 20 + 10  + 6; 100 is _ua tanaha ma tekau_, 2 × 40 + 10; 200 is _lima
tanaha_, 5 × 40; and 864,895 is 2 × 400,000 + 40,000 + 6 × 4000 + 2 × 400 +
2 × 40 + 10 + 5.[204] Such examples show that this secondary influence,
entering and incorporating itself as a part of a well-developed decimal
system, has radically changed it by the establishment of 4 as the primary
number base. The role which 10 now plays is peculiar. In the natural
formation of a quaternary scale new units would be introduced at 16, 64,
256, etc.; that is, at the square, the cube, and each successive power of
the base. But, instead of this, the new units are introduced at 10 × 4, 100
× 4, 1000 × 4, etc.; that is, at the products of 4 by each successive power
of the old base. This leaves the scale a decimal scale still, even while it
may justly be called quaternary; and produces one of the most singular and
interesting instances of number-system formation that has ever been
observed. In this connection it is worth noting that these Pacific island
number scales have been developed to very high limits--in some cases into
the millions. The numerals for these large numbers do not seem in any way
indefinite, but rather to convey to the mind of the native an idea as clear
as can well be conveyed by numbers of such magnitude. Beyond the limits
given, the islanders have indefinite expressions, but as far as can be
ascertained these are only used when the limits given above have actually
been passed. To quote one more example, the Hervey Islanders, who have a
binary-decimal scale, count as follows:

   5 kaviri (bunches of cocoanuts) = 1 takau = 20.
  10 takau                         = 1 rau   = 200.
  10 rau                           = 1 mano  = 2000.
  10 mano                          = 1 kiu   = 20,000.
  10 kiu                           = 1 tini  = 200,000.

Anything above this they speak of in an uncertain way, as _mano mano_ or
_tini tini_, which may, perhaps, be paralleled by our English phrases
"myriads upon myriads," and "millions of millions."[205] It is most
remarkable that the same quarter of the globe should present us with the
stunted number sense of the Australians, and, side by side with it, so
extended and intelligent an appreciation of numerical values as that
possessed by many of the lesser tribes of Polynesia.

The Luli of Paraguay[206] show a decided preference for the base 4. This
preference gives way only when they reach the number 10, which is an
ordinary digit numeral. All numbers above that point belong rather to
decimal than to quaternary numeration. Their numerals are:

   1.  alapea.
   2.  tamop.
   3.  tamlip.
   4.  lokep.
   5.  lokep moile alapea = 4 with 1,
        or is-alapea      = hand 1.
   6.  lokep moile tamop  = 4 with 2.
   7.  lokep moile tamlip = 4 with 3.
   8.  lokep moile lokep  = 4 with 4.
   9.  lokep moile lokep alapea = 4 with 4-1.
  10.  is yaoum = all the fingers of hand.
  11.  is yaoum moile alapea = all the fingers of hand with 1.
  20.  is elu yaoum = all the fingers of hand and foot.
  30.  is elu yaoum moile is-yaoum = all the fingers of hand and foot with
                                     all the fingers of hand.

Still another instance of quaternary counting, this time carrying with it a
suggestion of binary influence, is furnished by the Mocobi[207] of the
Parana region. Their scale is exceedingly rude, and they use the fingers
and toes almost exclusively in counting; only using their spoken numerals
when, for any reason, they wish to dispense with the aid of their hands and
feet. Their first eight numerals are:

   1.  iniateda.
   2.  inabaca.
   3.  inabacao caini   = 2 above.
   4.  inabacao cainiba = 2 above 2;
        or natolatata.
   5.  inibacao cainiba iniateda = 2 above 2-1;
        or natolatata iniateda   = 4-1.
   6.  natolatatata inibaca      = 4-2.
   7.  natolata inibacao-caini   = 4-2 above.
   8.  natolata-natolata         = 4-4.

There is probably no recorded instance of a number system formed on 6, 7,
8, or 9 as a base. No natural reason exists for the choice of any of these
numbers for such a purpose; and it is hardly conceivable that any race
should proceed beyond the unintelligent binary or quaternary stage, and
then begin the formation of a scale for counting with any other base than
one of the three natural bases to which allusion has already been made. Now
and then some anomalous fragment is found imbedded in an otherwise regular
system, which carries us back to the time when the savage was groping his
way onward in his attempt to give expression to some number greater than
any he had ever used before; and now and then one of these fragments is
such as to lead us to the border land of the might-have-been, and to cause
us to speculate on the possibility of so great a numerical curiosity as a
senary or a septenary scale. The Bretons call 18 _triouec'h_, 3-6, but
otherwise their language contains no hint of counting by sixes; and we are
left at perfect liberty to theorize at will on the existence of so unusual
a number word. Pott remarks[208] 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 tribe[211] call 6 _andee_, and 7 _anderee_, which
probably means 6-1. These words are to be regarded as accidental variations
on the ordinary laws of formation, and are no more significant of a desire
to count by sixes than is the Wallachian term _deu-maw_, which expresses 18
as 2-9, indicates the existence of a scale of which 9 is the base. One
remarkably interesting number system is that exhibited by the Mosquito
tribe[212] of Central America, who possess an extensive quinary-vigesimal
scale containing one binary and three senary compounds. The first ten words
of this singular scale, which has already been quoted, are:

   1.  kumi.
   2.  wal.
   3.  niupa.
   4.  wal-wal  = 2-2.
   5.  mata-sip = fingers of one hand.
   6.  matlalkabe.
   7.  matlalkabe pura kumi  = 6 + 1.
   8.  matlalkabe pura wal   = 6 + 2.
   9.  matlalkabe pura niupa = 6 + 3.
  10.  mata-wal-sip = fingers of the second hand.

In passing from 6 to 7, this tribe, also, has varied the almost universal
law of progression, and has called 7 6-1. Their 8 and 9 are formed in a
similar manner; but at 10 the ordinary method is resumed, and is continued
from that point onward. Few number systems contain as many as three
numerals which are associated with 6 as their base. In nearly all instances
we find such numerals singly, or at most in pairs; and in the structure of
any system as a whole, they are of no importance whatever. For example, in
the Pawnee, a pure decimal scale, we find the following odd sequence:[213]

   6.  shekshabish.
   7.  petkoshekshabish = 2-6, _i.e._ 2d 6.
   8.  touwetshabish    = 3-6, _i.e._ 3d 6.
   9.  loksherewa       = 10 - 1.

In the Uainuma scale the expressions for 7 and 8 are obviously referred to
6, though the meaning of 7 is not given, and it is impossible to guess what
it really does signify. The numerals in question are:[214]

   6.  aira-ettagapi.
   7.  aira-ettagapi-hairiwigani-apecapecapsi.
   8.  aira-ettagapi-matschahma = 6 + 2.

In the dialect of the Mille tribe a single trace of senary counting
appears, as the numerals given below show:[215]

   6.  dildjidji.
   7.  dildjidji me djuun = 6 + 1.

Finally, in the numerals used by the natives of the Marshall Islands, the
following curiously irregular sequence also contains a single senary
numeral:[216]

   6.  thil thino       = 3 + 3.
   7.  thilthilim-thuon = 6 + 1.
   8.  rua-li-dok       = 10 - 2.
   9.  ruathim-thuon    = 10 - 2 + 1.

Many years ago a statement appeared which at once attracted attention and
awakened curiosity. It was to the effect that the Maoris, the aboriginal
inhabitants of New Zealand, used as the basis of their numeral system the
number 11; and that the system was quite extensively developed, having
simple words for 121 and 1331, _i.e._ for the square and cube of 11. No
apparent reason existed for this anomaly, and the Maori scale was for a
long time looked upon as something quite exceptional and outside all
ordinary rules of number-system formation. But a closer and more accurate
knowledge of the Maori language and customs served to correct the mistake,
and to show that this system was a simple decimal system, and that the
error arose from the following habit. Sometimes when counting a number of
objects the Maoris would put aside 1 to represent each 10, and then those
so set aside would afterward be counted to ascertain the number of tens in
the heap. Early observers among this people, seeing them count 10 and then
set aside 1, at the same time pronouncing the word _tekau_, imagined that
this word meant 11, and that the ignorant savage was making use of this
number as his base. This misconception found its way into the early New
Zealand dictionary, but was corrected in later editions. It is here
mentioned only because of the wide diffusion of the error, and the interest
it has always excited.[217]

Aside from our common decimal scale, there exist in the English language
other methods of counting, some of them formal enough to be dignified by
the term _system_--as the sexagesimal method of measuring time and angular
magnitude; and the duodecimal system of reckoning, so extensively used in
buying and selling. Of these systems, other than decimal, two are noticed
by Tylor,[218] and commented on at some length, as follows:

"One is the well-known dicing set, _ace_, _deuce_, _tray_, _cater_,
_cinque_, _size_; thus _size-ace_ is 6-1, _cinques_ or _sinks_, double 5.
These came to us from France, and correspond with the common French
numerals, except _ace_, which is Latin _as_, a word of great philological
interest, meaning 'one.' The other borrowed set is to be found in the
_Slang Dictionary_. It appears that the English street-folk have adopted as
a means of secret communication a set of Italian numerals from the
organ-grinders and image-sellers, or by other ways through which Italian or
Lingua Franca is brought into the low neighbourhoods of London. In so doing
they have performed a philological operation not only curious but
instructive. By copying such expressions as _due soldi_, _tre soldi_, as
equivalent to 'twopence,' 'threepence,' the word _saltee_ became a
recognized slang term for 'penny'; and pence are reckoned as follows:

  oney saltee                              1d.  uno soldo.
  dooe saltee                              2d.  due soldi.
  tray saltee                              3d.  tre soldi.
  quarterer saltee                         4d.  quattro soldi.
  chinker saltee                           5d.  cinque soldi.
  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:

   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 producing a fractional result. With 8 as a base we
should obtain by successive halvings, 4, 2, 1. A similar process in our
decimal scale gives 5, 2-1/2, 1-1/4. All this is undeniably true, but,
granting the argument up to this point, one is then tempted to ask "What of
it?" A certain degree of simplicity would thereby be introduced into the
Theory of Numbers; but the only persons sufficiently interested in this
branch of mathematics to appreciate the benefit thus obtained are already
trained mathematicians, who are concerned rather with the pure science
involved, than with reckoning on any special base. A slightly increased
simplicity would appear in the work of stockbrokers, and others who reckon
extensively by quarters, eighths, and sixteenths. But such men experience
no difficulty whatever in performing their mental computations in the
decimal system; and they acquire through constant practice such quickness
and accuracy of calculation, that it is difficult to see how octonary
reckoning would materially assist them. Altogether, the reasons that have
in the past been adduced in favour of this form of arithmetic seem trivial.
There is no record of any tribe that ever counted by eights, nor is there
the slightest likelihood that such a system could ever meet with any
general favour. It is said that the ancient Saxons used the octonary
system,[220] but how, or for what purposes, is not stated. It is not to be
supposed that this was the common system of counting, for it is well known
that the decimal scale was in use as far back as the evidence of language
will take us. But the field of speculation into which one is led by the
octonary scale has proved most attractive to some, and the conclusion has
been soberly reached, that in the history of the Aryan race the octonary
was to be regarded as the predecessor of the decimal scale. In support of
this theory no direct evidence is brought forward, but certain verbal
resemblances. Those ignes fatuii of the philologist are made to perform
the duty of supporting an hypothesis which would never have existed but
for their own treacherous suggestions. Here is one of the most attractive
of them:

Between the Latin words _novus_, new, and _novem_, nine, there exists a
resemblance so close that it may well be more than accidental. Nine is,
then, the _new_ number; that is, the first number on a new count, of which
8 must originally have been the base. Pursuing this thought by
investigation into different languages, the same resemblance is found
there. Hence the theory is strengthened by corroborative evidence. In
language after language the same resemblance is found, until it seems
impossible to doubt, that in prehistoric times, 9 _was_ the new number--the
beginning of a second tale. The following table will show how widely spread
is this coincidence:

  Sanskrit,   navan = 9.          nava = new.
  Persian,    nuh = 9.            nau  = new.
  Greek,      [Greek: ennea] = 9. [Greek: neos] = new.
  Latin,      novem = 9.          novus = new.
  German,     neun = 9.           neu  = new.
  Swedish,    nio = 9.            ny   = new.
  Dutch,      negen = 9.          nieuw = new.
  Danish,     ni = 9.             ny   = new.
  Icelandic,  nyr = 9.            niu  = new.
  English,    nine = 9.           new  = new.
  French,     neuf = 9.           nouveau = new.
  Spanish,    nueve = 9.          neuvo = new.
  Italian,    nove = 9.           nuovo = new.
  Portuguese, nove = 9.           novo = new.
  Irish,      naoi = 9.           nus  = new.
  Welsh,      naw = 9.            newydd = new.
  Breton,     nevez = 9.          nuhue = new.[221]

This table might be extended still further, but the above examples show how
widely diffused throughout the Aryan languages is this resemblance. The
list certainly is an impressive one, and the student is at first thought
tempted to ask whether all these resemblances can possibly have been
accidental. But a single consideration sweeps away the entire argument as
though it were a cobweb. All the languages through which this verbal
likeness runs are derived directly or indirectly from one common stock; and
the common every-day words, "nine" and "new," have been transmitted from
that primitive tongue into all these linguistic offspring with but little
change. Not only are the two words in question akin in each individual
language, but _they are akin in all the languages_. Hence all these
resemblances reduce to a single resemblance, or perhaps identity, that
between the Aryan words for "nine" and "new." This was probably an
accidental resemblance, no more significant than any one of the scores of
other similar cases occurring in every language. If there were any further
evidence of the former existence of an Aryan octonary scale, the
coincidence would possess a certain degree of significance; but not a shred
has ever been produced which is worthy of consideration. If our remote
ancestors ever counted by eights, we are entirely ignorant of the fact, and
must remain so until much more is known of their language than scholars now
have at their command. The word resemblances noted above are hardly more
significant than those occurring in two Polynesian languages, the Fatuhivan
and the Nakuhivan,[222] where "new" is associated with the number 7. In the
former case 7 is _fitu_, and "new" is _fou_; in the latter 7 is _hitu_, and
"new" is _hou_. But no one has, because of this likeness, ever suggested
that these tribes ever counted by the senary method. Another equally
trivial resemblance occurs in the Tawgy and the Kamassin languages,[223]
thus:


  TAWGY.                               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 8
was the number base used.

In a recent antiquarian work of considerable interest, the author examines
into the question of a former octonary system of counting among the various
races of the world, particularly those of Asia, and brings to light much
curious and entertaining material respecting the use of this number. Its
use and importance in China, India, and central Asia, as well as among some
of the islands of the Pacific, and in Central America, leads him to the
conclusion that there was a time, long before the beginning of recorded
history, when 8 was the common number base of the world. But his conclusion
has no basis in his own material even. The argument cannot be examined
here, but any one who cares to investigate it can find there an excellent
illustration of the fact that a pet theory may take complete possession of
its originator, and reduce him finally to a state of infantile
subjugation.[224]

Of all numbers upon which a system could be based, 12 seems to combine in
itself the greatest number of advantages. It is capable of division by 2,
3, 4, and 6, and hence admits of the taking of halves, thirds, quarters,
and sixths of itself without the introduction of fractions in the result.
From a commercial stand-point this advantage is very great; so great that
many have seriously advocated the entire abolition of the decimal scale,
and the substitution of the duodecimal in its stead. It is said that
Charles XII. of Sweden was actually contemplating such a change in his
dominions at the time of his death. In pursuance of this idea, some writers
have gone so far as to suggest symbols for 10 and 11, and to recast our
entire numeral nomenclature to conform to the duodecimal base.[225] Were
such a change made, we should express the first nine numbers as at present,
10 and 11 by new, single symbols, and 12 by 10. From this point the
progression would be regular, as in the decimal scale--only the same
combination of figures in the different scales would mean very different
things. Thus, 17 in the decimal scale would become 15 in the duodecimal;
144 in the decimal would become 100 in the duodecimal; and 1728, the cube
of the new base, would of course be represented by the figures 1000.

It is impossible that any such change can ever meet with general or even
partial favour, so firmly has the decimal scale become intrenched in its
position. But it is more than probable that a large part of the world of
trade and commerce will continue to buy and sell by the dozen, the gross,
or some multiple or fraction of the one or the other, as long as buying and
selling shall continue. Such has been its custom for centuries, and such
will doubtless be its custom for centuries to come. The duodecimal is not a
natural scale in the same sense as are the quinary, the decimal, and the
vigesimal; but it is a system which is called into being long after the
complete development of one of the natural systems, solely because of the
simple and familiar fractions into which its base is divided. It is the
scale of civilization, just as the three common scales are the scales of
nature. But an example of its use was long sought for in vain among the
primitive races of the world. Humboldt, in commenting on the number systems
of the various peoples he had visited during his travels, remarked that no
race had ever used exclusively that best of bases, 12. But it has recently
been announced[226] 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.





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 which were, and in
many cases still are, in daily use among the uncivilized races of mankind.

In its structure the quinary is the simplest, the most primitive, of the
natural systems. Its base is almost always expressed by a word meaning
"hand," or by some equivalent circumlocution, and its digital origin is
usually traced without difficulty. A consistent formation would require the
expression of 10 by some phrase meaning "two fives," 15 by "three fives,"
etc. Such a scale is the one obtained from the Betoya language, already
mentioned in Chapter III., where the formation of the numerals is purely
quinary, as the following indicate:[227]

   5.  teente = 1 hand.
  10.  cayaente, or caya huena = 2 hands.
  15.  toazumba-ente = 3 hands.
  20.  caesa-ente    = 4 hands.

The same formation appears, with greater or less distinctness, in many of
the quinary scales already quoted, and in many more of which mention might
be made. Collecting the significant numerals from a few such scales, and
tabulating them for the sake of convenience of comparison, we see this
point clearly illustrated by the following:


  TAMANAC.

   5.  amnaitone = 1 hand.
  10.  amna atse ponare = 2 hands.


  ARAWAK, GUIANA.

   5.  abba tekkabe  = 1 hand.
  10.  biamantekkabe = 2 hands.


  JIVIRO.

   5.  alacötegladu = 1 hand.
  10.  catögladu    = 2 hands.


  NIAM NIAM

   5.  biswe
  10.  bauwe = 2d 5.


  NENGONES

   5.  se dono       = the end (of the fingers of 1 hand).
  10.  rewe tubenine = 2 series (of fingers).


  SESAKE.[228]

   5.  lima     = hand.
  10.  dua lima = 2 hands.


  AMBRYM.[229]

   5.  lim    = hand.
  10.  ra-lim = 2 hands.


  PAMA.[229]

   5.  e-lime     = hand.
  10.  ha-lua-lim = the 2 hands.


  DINKA.[230]

   5.  wdyets.
  10.  wtyer, or wtyar = 5 × 2.


  BARI

   5.  kanat
  10.  puök = 5 + 5?


  KANURI

   5.  ugu.
  10.  megu = 2 × 5.


  RIO NORTE AND SAN ANTONIO.[231]

   5.  juyopamauj.
  10.  juyopamauj ajte = 5 × 2.


  API.[232]

   5.  lima.
  10.  lua-lima = 2 × 5.


  ERROMANGO

   5.  suku-rim.
  10.  nduru-lim = 2 × 5.


  TLINGIT, BRITISH COLUMBIA.[233]

   5.  kedjin (from djin = hand).
  10.  djinkat = both hands?

Thus far the quinary formation is simple and regular; and in view of the
evidence with which these and similar illustrations furnish us, it is most
surprising to find an eminent authority making the unequivocal statement
that the number 10 is nowhere expressed by 2 fives[234]--that all tribes
which begin their count on a quinary base express 10 by a simple word. It
is a fact, as will be fully illustrated in the following pages, that
quinary number systems, when extended, usually merge into either the
decimal or the vigesimal. The result is, of course, a compound of two, and
sometimes of three, systems in one scale. A pure quinary or vigesimal
number system is exceedingly rare; but quinary scales certainly do exist in
which, as far as we possess the numerals, no trace of any other influence
appears. It is also to be noticed that some tribes, like the Eskimos of
Point Barrow, though their systems may properly be classed as mixed
systems, exhibit a decided preference for 5 as a base, and in counting
objects, divided into groups of 5, obtaining the sum in this way.[235]

But the savage, after counting up to 10, often finds himself unconsciously
impelled to depart from his strict reckoning by fives, and to assume a new
basis of reference. Take, for example, the Zuñi system, in which the first
2 fives are:

   5.  öpte = the notched off.
  10.  astem'thla = all the fingers.

It will be noticed that the Zuñi does not say "two hands," or "the fingers
of both hands," but simply "all the fingers." The 5 is no longer prominent,
but instead the mere notion of one entire count of the fingers has taken
its place. The division of the fingers into two sets of five each is still
in his mind, but it is no longer the leading idea. As the count proceeds
further, the quinary base may be retained, or it may be supplanted by a
decimal or a vigesimal base. How readily the one or the other may
predominate is seen by a glance at the following numerals:


  GALIBI.[236]

   5.  atoneigne oietonaï = 1 hand.
  10.  oia batoue = the other hand.
  20.  poupoupatoret oupoume = feet and hands.
  40.  opoupoume = twice the feet and hands.


  GUARANI.[237]

   5.  ace popetei    = 1 hand.
  10.  ace pomocoi    = 2 hands.
  20.  acepo acepiabe = hands and feet.


  FATE.[238]

   5.  lima       = hand.
  10.  relima     = 2 hands.
  20.  relima rua = (2 × 5) × 2.


  KIRIRI

   5.  mibika misa      = 1 hand.
  10.  mikriba misa sai = both hands.
  20.  mikriba nusa ideko ibi sai = both hands together with the feet.


  ZAMUCO

   5.  tsuena yimana-ite = ended 1 hand.
  10.  tsuena yimana-die = ended both hands.
  20.  tsuena yiri-die   = ended both feet.


  PIKUMBUL

   5.  mulanbu.
  10.  bularin murra = belonging to the two hands.
  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 both are
used. Wherever this is done, the greater of the two bases is always to be
regarded as the principal number base of the language, and the 5 as
entirely subordinate to it. It is hardly correct to say that, as a number
system is extended, the quinary element disappears and gives place to the
decimal or vigesimal, but rather that it becomes a factor of quite
secondary importance in the development of the scale. If, for example, 8 is
expressed by 5-3 in a quinary decimal system, 98 will be 9 × 10 + 5-3. The
quinary element does not disappear, but merely sinks into a relatively
unimportant position.

One of the purest examples of quinary numeration is that furnished by the
Betoya scale, already given in full in Chapter III., and briefly mentioned
at the beginning of this chapter. In the simplicity and regularity of its
construction it is so noteworthy that it is worth repeating, as the first
of the long list of quinary systems given in the following pages. No
further comment is needed on it than that already made in connection with
its digital significance. As far as given by Dr. Brinton the scale is:

   1.  tey.
   2.  cayapa.
   3.  toazumba.
   4.  cajezea     = 2 with plural termination.
   5.  teente      = hand.
   6.  teyente tey = hand 1.
   7.  teyente cayapa    = hand 2.
   8.  teyente toazumba  = hand 3.
   9.  teyente caesea    = hand 4.
  10.  caya ente, or caya huena = 2 hands.
  11.  caya ente-tey     = 2 hands 1.
  15.  toazumba-ente     = 3 hands.
  16.  toazumba-ente-tey = 3 hands 1.
  20.  caesea ente       = 4 hands.

A far more common method of progression is furnished by languages which
interrupt the quinary formation at 10, and express that number by a single
word. Any scale in which this takes place can, from this point onward, be
quinary only in the subordinate sense to which allusion has just been made.
Examples of this are furnished in a more or less perfect manner by nearly
all so-called quinary-vigesimal and quinary-decimal scales. As fairly
representing this phase of number-system structure, I have selected the
first 20 numerals from the following languages:


  WELSH.[240]

   1.  un.
   2.  dau.
   3.  tri.
   4.  pedwar.
   5.  pump.
   6.  chwech.
   7.  saith.
   8.  wyth.
   9.  naw.
  10.  deg.
  11.  un ar ddeg     = 1 + 10.
  12.  deuddeg        = 2 + 10.
  13.  tri ar ddeg    = 3 + 10.
  14.  pedwar ar ddeg = 4 + 10.
  15.  pymtheg        = 5 + 10.
  16.  un ar bymtheg  = 1 + 5 + 10.
  17.  dau ar bymtheg = 2 + 5 + 10.
  18.  tri ar bymtheg = 3 + 5 + 10.
  19.  pedwar ar bymtheg = 4 + 5 + 10.
  20.  ugain.


  NAHUATL.[241]

   1.  ce.
   2.  ome.
   3.  yei.
   4.  naui.
   5.  macuilli.
   6.  chiquacen = [5] + 1.
   7.  chicome   = [5] + 2.
   8.  chicuey   = [5] + 3.
   9.  chiucnaui = [5] + 4.
  10.  matlactli.
  11.  matlactli oce    = 10 + 1.
  12.  matlactli omome  = 10 + 2.
  13.  matlactli omey   = 10 + 3.
  14.  matlactli onnaui = 10 + 4.
  15.  caxtolli.
  16.  caxtolli oce    = 15 + 1.
  17.  caxtolli omome  = 15 + 2.
  18.  caxtolli omey   = 15 + 3.
  19.  caxtolli onnaui = 15 + 4.
  20.  cempualli = 1 account.


  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 often found in primitive systems, they have, at a venture, been
given here.


  LIFU, LOYALTY ISLANDS.[244]

   1.  ca.
   2.  lue.
   3.  koeni.
   4.  eke.
   5.  tji pi.
   6.  ca ngemen    = 1 above.
   7.  lue ngemen   = 2 above.
   8.  koeni ngemen = 3 above.
   9.  eke ngemen   = 4 above.
  10.  lue pi       = 2 × 5.
  11.  ca ko.
  12.  lue ko.
  13.  koeni ko.
  14.  eke ko.
  15.  koeni pi = 3 × 5.
  16.  ca huai ano.
  17.  lua huai ano.
  18.  koeni huai ano.
  19.  eke huai ano.
  20.  ca atj = 1 man.


  BONGO.[245]

   1.  kotu.
   2.  ngorr.
   3.  motta.
   4.  neheo.
   5.  mui.
   6.  dokotu  = [5] + 1.
   7.  dongorr = [5] + 2.
   8.  domotta = [5] + 3.
   9.  doheo   = [5] + 4.
  10.  kih.
  11.  ki dokpo kotu  = 10 + 1.
  12.  ki dokpo ngorr = 10 + 2.
  13.  ki dokpo motta = 10 + 3.
  14.  ki dokpo neheo = 10 + 4.
  15.  ki dokpo mui   = 10 + 5.
  16.  ki dokpo mui do mui okpo kotu  = 10 + 5 more, to 5, 1 more.
  17.  ki dokpo mui do mui okpo ngorr = 10 + 5 more, to 5, 2 more.
  18.  ki dokpo mui do mui okpo motta = 10 + 5 more, to 5, 3 more.
  19.  ki dokpo mui do mui okpo nehea = 10 + 5 more, to 5, 4 more.
  20.  mbaba kotu.

Above 20, the Lufu and the Bongo systems are vigesimal, so that they are,
as a whole, mixed systems.

The Welsh scale begins as though it were to present a pure decimal
structure, and no hint of the quinary element appears until it has passed
15. The Nahuatl, on the other hand, counts from 5 to 10 by the ordinary
quinary method, and then appears to pass into the decimal form. But when 16
is reached, we find the quinary influence still persistent; and from this
point to 20, the numeral words in both scales are such as to show that the
notion of counting by fives is quite as prominent as the notion of
referring to 10 as a base. Above 20 the systems become vigesimal, with a
quinary or decimal structure appearing in all numerals except multiples of
20. Thus, in Welsh, 36 is _unarbymtheg ar ugain_, 1 + 5  + 10 + 20; and in
Nahuatl the same number is _cempualli caxtolli oce_, 20 + 15 + 1. Hence
these and similar number systems, though commonly alluded to as vigesimal,
are really mixed scales, with 20 as their primary base. The Canaque scale
differs from the Nahuatl only in forming a compound word for 15, instead of
introducing a new and simple term.

In the examples which follow, it is not thought best to extend the lists of
numerals beyond 10, except in special instances where the illustration of
some particular point may demand it. The usual quinary scale will be found,
with a few exceptions like those just instanced, to have the following
structure or one similar to it in all essential details: 1, 2, 3, 4, 5,
5-1, 5-2, 5-3, 5-4, 10, 10-1, 10-2, 10-3, 10-4, 10-5, 10-5-1, 10-5-2,
10-5-3, 10-5-4, 20. From these forms the entire system can readily be
constructed as soon as it is known whether its principal base is to be 10
or 20.

Turning first to the native African languages, I have selected the
following quinary scales from the abundant material that has been collected
by the various explorers of the "Dark Continent." In some cases the
numerals of certain tribes, as given by one writer, are found to differ
widely from the same numerals as reported by another. No attempt has been
made at comparison of these varying forms of orthography, which are usually
to be ascribed to difference of nationality on the part of the collectors.


  FELOOPS.[246]

   1.  enory.
   2.  sickaba, or cookaba.
   3.  sisajee.
   4.  sibakeer.
   5.  footuck.
   6.  footuck-enory    = 5-1.
   7.  footuck-cookaba  = 5-2.
   8.  footuck-sisajee  = 5-3.
   9.  footuck-sibakeer = 5-4.
  10.  sibankonyen.


  KISSI.[247]

   1.  pili.
   2.  miu.
   3.  nga.
   4.  iol.
   5.  nguenu.
   6.  ngom-pum   = 5-1.
   7.  ngom-miu   = 5-2.
   8.  ngommag    = 5-3.
   9.  nguenu-iol = 5-4.
  10.  to.


  ASHANTEE.[248]

   1.  tah.
   2.  noo.
   3.  sah.
   4.  nah.
   5.  taw.
   6.  torata      = 5 + 1.
   7.  toorifeenoo = 5 + 2.
   8.  toorifeessa = 5 + 3.
   9.  toorifeena  = 5 + 4.
  10.  nopnoo.


  BASA.[249]

   1.  do.
   2.  so.
   3.  ta.
   4.  hinye.
   5.  hum.
   6.  hum-le-do    = 5 + 1.
   7.  hum-le-so    = 5 + 2.
   8.  hum-le-ta    = 5 + 3.
   9.  hum-le-hinyo = 5 + 4.
  10.  bla-bue.


  JALLONKAS.[250]

   1.  kidding.
   2.  fidding.
   3.  sarra.
   4.  nani.
   5.  soolo.
   6.  seni.
   7.  soolo ma fidding = 5 + 2.
   8.  soolo ma sarra   = 5 + 3.
   9.  soolo ma nani    = 5 + 4.
  10.  nuff.


  KRU.

   1.  da-do.
   2.  de-son.
   3.  de-tan.
   4.  de-nie.
   5.  de-mu.
   6.  dme-du = 5-1.
   7.  ne-son = [5] + 2.
   8.  ne-tan = [5] + 3.
   9.  sepadu = 10 - 1?
  10.  pua.


  JALOFFS.[251]

   1.  wean.
   2.  yar.
   3.  yat.
   4.  yanet.
   5.  judom.
   6.  judom-wean  = 5-1.
   7.  judom-yar   = 5-2.
   8.  judom-yat   = 5-3.
   9.  judom yanet = 5-4.
  10.  fook.


  GOLO.[252]

   1.  mbali.
   2.  bisi.
   3.  bitta.
   4.  banda.
   5.  zonno.
   6.  tsimmi tongbali = 5 + 1.
   7.  tsimmi tobisi   = 5 + 2.
   8.  tsimmi tobitta  = 5 + 3.
   9.  tsimmi to banda = 5 + 4.
  10.  nifo.


  FOULAH.[253]

   1.  go.
   2.  deeddee.
   3.  tettee.
   4.  nee.
   5.  jouee.
   6.  jego      = 5-1.
   7.  jedeeddee = 5-2.
   8.  je-tettee = 5-3.
   9.  je-nee    = 5-4.
  10.  sappo.


  SOUSSOU.[254]

   1.  keren.
   2.  firing.
   3.  sarkan.
   4.  nani.
   5.  souli.
   6.  seni.
   7.  solo-fere     = 5-2.
   8.  solo-mazarkan = 5 + 3.
   9.  solo-manani   = 5 + 4.
  10.  fu.


  BULLOM.[255]

   1.  bul.
   2.  tin.
   3.  ra.
   4.  hyul.
   5.  men.
   6.  men-bul  = 5-1.
   7.  men-tin  = 5-2.
   8.  men-ra   = 5-3.
   9.  men-hyul = 5-4.
  10.  won.


  VEI.[256]

   1.  dondo.
   2.  fera.
   3.  sagba.
   4.  nani.
   5.  soru.
   6.  sun-dondo = 5-1.
   7.  sum-fera  = 5-2.
   8.  sun-sagba = 5-3.
   9.  sun-nani  = 5-4.
  10.  tan.


  DINKA.[257]

   1.  tok.
   2.  rou.
   3.  dyak.
   4.  nuan.
   5.  wdyets.
   6.  wdetem   = 5-1.
   7.  wderou   = 5-2.
   8.  bet, bed = 5-3.
   9.  wdenuan  = 5-4.
  10.  wtyer    = 5 × 2.


  TEMNE.

   1.  in.
   2.  ran.
   3.  sas.
   4.  anle.
   5.  tr-amat.
   6.  tr-amat rok-in    = 5 + 1.
   7.  tr-amat de ran    = 5 + 2.
   8.  tr-amat re sas    = 5 + 3.
   9.  tr-amat ro n-anle = 5 + 4.
  10.  tr-ofatr.


  ABAKER.[258]

   1.  kili.
   2.  bore.
   3.  dotla.
   4.  ashe.
   5.  ini.
   6.  im kili   = 5-1.
   7.  im-bone   = 5-2.
   8.  ini-dotta = 5-3.
   9.  tin ashe  = 5-4.
  10.  chica.


  BAGRIMMA.[259]

   1.  kede.
   2.  sab.
   3.  muta.
   4.  so.
   5.  mi.
   6.  mi-ga = 5 + 1.
   7.  tsidi.
   8.  marta = 5 + 2.
   9.  do-so = [5] + 3
  10.  duk-keme.


  PAPAA.[260]

   1.  depoo.
   2.  auwi.
   3.  ottong.
   4.  enne.
   5.  attong.
   6.  attugo.
   7.  atjuwe    = [5] + 2.
   8.  attiatong = [5] + 3.
   9.  atjeenne  = [5] + 4.
  10.  awo.


  EFIK.[261]

   1.  kiet.
   2.  iba.
   3.  ita.
   4.  inan.
   5.  itiun.
   6.  itio-kiet = 5-1.
   7.  itia-ba   = 5-2.
   8.  itia-eta  = 5-3.
   9.  osu-kiet  = 10 - 1?
  10.  duup.


  NUPE.[262]

   1.  nini.
   2.  gu-ba.
   3.  gu-ta.
   4.  gu-ni.
   5.  gu-tsun.
   6.  gu-sua-yin = 5 + 1.
   7.  gu-tua-ba  = 5 + 2.
   8.  gu-tu-ta   = 5 + 3.
   9.  gu-tua-ni  = 5 + 4.
  10.  gu-wo.


  MOKKO.[263]

   1.  kiä.
   2.  iba.
   3.  itta.
   4.  inan.
   5.  üttin.
   6.  itjüekee = 5 + 1.
   7.  ittiaba  = 5 + 2.
   8.  itteiata = 5 + 3.
   9.  huschukiet.
  10.  büb.


  KANURI.[264]

   1.  tilo.
   2.  ndi.
   3.  yasge.
   4.  dege.
   5.  ugu.
   6.  arasge = 5 + 1.
   7.  tulur.
   8.  wusge  = 5 + 3.
   9.  legar.
  10.  megu   = 2 × 5.


  BININ.[265]

   1.  bo.
   2.  be.
   3.  la.
   4.  nin.
   5.  tang.
   6.  tahu = 5 + 1?
   7.  tabi = 5 + 2.
   8.  tara = 5 + 3.
   9.  ianin (tanin?) = 5 + 4?
  10.  te.


  KREDY.[266]

   1.  baia.
   2.  rommu.
   3.  totto.
   4.  sosso.
   5.  saya.
   6.  yembobaia  = [5] + 1.
   7.  yemborommu = [5] + 2.
   8.  yembototto = [5] + 3.
   9.  yembososso = [5] + 4.
  10.  puh.


  HERERO.[267]

   1.  mue.
   2.  vari.
   3.  tatu.
   4.  ne.
   5.  tano.
   6.  hambou-mue  = [5] + 1.
   7.  hambou-vari = [5] + 2.
   8.  hambou-tatu = [5] + 3.
   9.  hambou-ne   = [5] + 4.
  10.


  KI-YAU.[268]

   1.  jumo.
   2.  wawiri.
   3.  watatu.
   4.  mcheche.
   5.  msano.
   6.  musano na jumo    = 5 + 1.
   7.  musano na wiri    = 5 + 2.
   8.  musano na watatu  = 5 + 3.
   9.  musano na mcheche = 5 + 4.
  10.  ikumi.


  FERNANDO PO.[269]

   1.  muli.
   2.  mempa.
   3.  meta.
   4.  miene.
   5.  mimito.
   6.  mimito na muli  = 5 + 1.
   7.  mimito na mempa = 5 + 2.
   8.  mimito na meta  = 5 + 3.
   9.  mimito na miene = 5 + 4.
  10.  miemieu = 5-5?


  KI-NYASSA

   1.  kimodzi.
   2.  vi-wiri.
   3.  vi-tatu.
   4.  vinye.
   5.  visano.
   6.  visano na kimodzi = 5 + 1.
   7.  visano na vi-wiri = 5 + 2.
   8.  visano na vitatu  = 5 + 3.
   9.  visano na vinye   = 5 + 4.
  10.  chikumi.


  BALENGUE.[270]

   1.  guevoho.
   2.  ibare.
   3.  raro.
   4.  inaï.
   5.  itano.
   6.  itano na guevoho = 5 + 1.
   7.  itano na ibare   = 5 + 2.
   8.  itano na raro    = 5 + 3.
   9.  itano na inaï    = 5 + 4.
  10.  ndioum, or nai-hinaï.


  KUNAMA.[271]

   1.  ella.
   2.  bare.
   3.  sadde.
   4.  salle.
   5.  kussume.
   6.  kon-t'-ella  = hand 1.
   7.  kon-te-bare  = hand 2.
   8.  kon-te-sadde = hand 3.
   9.  kon-te-salle = hand 4.
  10.  kol-lakada.


  GOLA.[272]

   1.  ngoumou.
   2.  ntie.
   3.  ntaï.
   4.  tina.
   5.  nonon.
   6.  diegoum = [5] + 1.
   7.  dientie = [5] + 2.
   8.  dietai  = [5] + 3.
   9.  dectina = [5] + 4.
  10.  esia.


  BAREA.[273]

   1.  doko
   2.  arega.
   3.  sane.
   4.  sone.
   5.  oita.
   6.  data.
   7.  dz-ariga = 5 + 2.
   8.  dis-sena = 5 + 3.
   9.  lefete-mada = without 10.
  10.  lefek.


  MATIBANI.[274]

   1.  mosa.
   2.  pili.
   3.  taru.
   4.  teje.
   5.  taru.
   6.  tana mosa = 5-1.
   7.  tana pili = 5-2.
   8.  tana taru = 5-3.
   9.  loco.
  10.  loco nakege.


  BONZÉ.[275]

   1.  tan.
   2.  vele.
   3.  daba.
   4.  nani.
   5.  lolou.
   6.  maïda    = [5] + 1.
   7.  maïfile  = [5] + 2.
   8.  maïshaba = [5] + 3.
   9.  maïnan   = [5] + 4.
  10.  bou.


  MPOVI

   1.  moueta.
   2.  bevali.
   3.  betata.
   4.  benaï.
   5.  betani.
   6.  betani moueta = 5-1.
   7.  betani bevali = 5-2.
   8.  betani betata = 5-3.
   9.  betani benai  = 5-4.
  10.  nchinia.


  TRITON'S BAY, NEW QUINEA.[276]

   1.  samosi.
   2.  roueti.
   3.  tourou.
   4.  faat.
   5.  rimi.
   6.  rim-samosi = 5-1.
   7.  rim-roueti = 5-2.
   8.  rim-tourou = 5-3.
   9.  rim-faat   = 5-4.
  10.  outsia.


  ENDE, OR FLORES.[277]

   1.  sa.
   2.  zua.
   3.  telu.
   4.  wutu.
   5.  lima     = hand.
   6.  lima-sa  = 5-1, or hand 1.
   7.  lima-zua = 5-2.
   8.  rua-butu = 2 × 4?
   9.  trasa    = [10] - 1?
  10.  sabulu.


  MALLICOLO.[278]

   1.  tseekaee.
   2.  ery.
   3.  erei.
   4.  ebats.
   5.  ereem.
   6.  tsookaee = [5] + 1.
   7.  gooy     = [5] + 2.
   8.  hoorey   = [5] + 3.
   9.  goodbats = [5] + 4.
  10.  senearn.


  EBON, MARSHALL ISLANDS.[279]

   1.  iuwun.
   2.  drud.
   3.  chilu.
   4.  emer.
   5.  lailem.
   6.  chilchinu   = 5 + 1.
   7.  chilchime   = 5 + 2.
   8.  twalithuk   = [10] - 2.
   9.  twahmejuwou = [10] - 1.
  10.  iungou.


  UEA, 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.


  NEW CALEDONIA.[283]

   1.  parai.
   2.  paroo.
   3.  parghen.
   4.  parbai.
   5.  panim.
   6.  panim-gha  = 5-1.
   7.  panim-roo  = 5-2.
   8.  panim-ghen = 5-3.
   9.  panim-bai  = 5-4.
  10.  parooneek.


  YENGEN, NEW CAL.[284]

   1.  hets.
   2.  heluk.
   3.  heyen.
   4.  pobits.
   5.  nim       = hand.
   6.  nim-wet   = 5-1.
   7.  nim-weluk = 5-2.
   8.  nim-weyen = 5-3.
   9.  nim-pobit = 5-4.
  10.  pain-duk.


  ANEITEUM.[285]

   1.  ethi.
   2.  ero.
   3.  eseik.
   4.  manohwan.
   5.  nikman.
   6.  nikman cled et ethi  = 5 + 1.
   7.  nikman cled et oro   = 5 + 2.
   8.  nikman cled et eseik = 5 + 3.
   9.  nikman cled et manohwan = 5 + 4.
  10.  nikman lep ikman     = 5 + 5.


  TANNA

   1.  riti.
   2.  karu.
   3.  kahar.
   4.  kefa.
   5.  krirum.
   6.  krirum riti   = 5-1.
   7.  krirum karu   = 5-2.
   8.  krirum kahar? = 5-3.
   9.  krirum kefa?  = 5-4.
  10.  ----


  EROMANGA

   1.  sai.
   2.  duru.
   3.  disil.
   4.  divat.
   5.  siklim  = 1 hand.
   6.  misikai = other 1?
   7.  siklim naru     = 5-2.
   8.  siklim disil    = 5-3.
   9.  siklim mindivat = 5 + 4.
  10.  narolim = 2 hands.


  FATE, NEW HEB.[286]

   1.  iskei.
   2.  rua.
   3.  tolu.
   4.  bate.
   5.  lima    = hand.
   6.  la tesa = other 1.
   7.  la rua  = other 2.
   8.  la tolu = other 3.
   9.  la fiti = other 4.
  10.  relima  = 2 hands.


  API, NEW HEB.

   1.  tai.
   2.  lua.
   3.  tolu.
   4.  vari.
   5.  lima     = hand.
   6.  o rai    = other 1.
   7.  o lua    = other 2.
   8.  o tolo   = other 3.
   9.  o vari   = other 4.
  10.  lua lima = 2 hands.


  SESAKE, NEW HEB.

   1.  sikai.
   2.  dua.
   3.  dolu.
   4.  pati.
   5.  lima     = hand.
   6.  la tesa  = other 1.
   7.  la dua   = other 2.
   8.  la dolu  = other 3.
   9.  lo veti  = other 4.
  10.  dua lima = 2 hands.


  PAMA, NEW HEB.

   1.  tai.
   2.  e lua.
   3.  e tolu.
   4.  e hati.
   5.  e lime  = hand.
   6.  a hitai = other 1.
   7.  o lu    = other 2.
   8.  o tolu  = other 3.
   9.  o hati  = other 4.
  10.  ha lua lim = 2 hands


  AURORA, NEW HEB.

   1.  tewa.
   2.  i rua.
   3.  i tol.
   4.  i vat.
   5.  tavalima = 1 hand.
   6.  lava tea = other 1.
   7.  lava rua = other 2.
   8.  lava tol = other 3.
   9.  la vat   = other 4.
  10.  sanwulu  = two sets.


  TOBI.[287]

   1.  yat.
   2.  glu.
   3.  ya.
   4.  uan.
   5.  yanim = 1 hand.
   6.  yawor = other 1.
   7.  yavic = other 2.
   8.  yawa  = other 3.
   9.  yatu  = other 4.
  10.  yasec.


  PALM ISLAND.[288]

   1.  yonkol.
   2.  yakka.
   3.  tetjora.
   4.  tarko.
   5.  yonkol mala = 1 hand.


  JAJOWERONG, VICTORIA.[288]

   1.  kiarp.
   2.  bulaits.
   3.  bulaits kiarp   = 2-1.
   4.  bulaits bulaits = 2-2.
   5.  kiarp munnar    = 1 hand.
   6.  bulaits bulaits bulaits = 2-2-2.
  10.  bulaits munnar  = 2 hands.

The last two scales deserve special notice. They are Australian scales, and
the former is strongly binary, as are so many others of that continent. But
both show an incipient quinary tendency in their names for 5 and 10.


  CAMBODIA.[289]

   1.  muy.
   2.  pir.
   3.  bey.
   4.  buon.
   5.  pram.
   6.  pram muy  = 5-1.
   7.  pram pil  = 5-2.
   8.  pram bey  = 5-3.
   9.  pram buon = 5-4.
  10.  dap.


  TSCHUKSCHI.[290]

   1.  inen.
   2.  nirach.
   3.  n'roch.
   4.  n'rach.
   5.  miligen = hand.
   6.  inen miligen   = 1-5.
   7.  nirach miligen = 2-5.
   8.  anwrotkin.
   9.  chona tsinki.
  10.  migitken = both hands.


  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.


  ALEUTS[294]

   1.  ataqan.
   2.  aljak.
   3.  qankun.
   4.  sitsin.
   5.  tsan    = my hand.
   6.  atun    = 1 + 5.
   7.  ulun    = 2 + 5.
   8.  qamtsin = 3 + 5.
   9.  sitsin  = 4 + 5.
  10.  hatsiq.


  TCHIGLIT, MACKENZIE R.[295]

   1.  ataotçirkr.
   2.  aypak, or malloerok.
   3.  illaak, or piñatcut.
   4.  tçitamat.
   5.  tallemat.
   6.  arveneloerit.
   7.  arveneloerit-aypak    = 5 + 2.
   8.  arveneloerit-illaak   = 5 + 3.
   9.  arveneloerit-tçitamat = 5 + 4.
  10.  krolit.


  SAHAPTIN (NEZ PERCES).[296]

   1.  naks.
   2.  lapit.
   3.  mitat.
   4.  pi-lapt  = 2 × 2.
   5.  pachat.
   6.  oi-laks  = [5] + 1.
   7.  oi-napt  = [5] + 2.
   8.  oi-matat = [5] + 3.
   9.  koits.
  10.  putimpt.


  GREENLAND.[297]

   1.  atauseq.
   2.  machdluq.
   3.  pinasut.
   4.  sisamat
   5.  tadlimat.
   6.  achfineq-atauseq  = other hand 1.
   7.  achfineq-machdluq = other hand 2.
   8.  achfineq-pinasut  = other hand 3.
   9.  achfineq-sisamat  = other hand 4.
  10.  qulit.
  11.  achqaneq-atauseq  = first foot 1.
  12.  achqaneq-machdluq = first foot 2.
  13.  achqaneq-pinasut  = first foot 3.
  14.  achqaneq-sisamat  = first foot 4.
  15.  achfechsaneq?
  16.  achfechsaneq-atauseq  = other foot 1.
  17.  achfechsaneq-machdlup = other foot 2.
  18.  achfechsaneq-pinasut  = other foot 3.
  19.  achfechsaneq-sisamat  = other foot 4.
  20.  inuk navdlucho    = a man ended.

Up to this point the Greenlander's scale is almost purely quinary. Like
those of which mention was made at the beginning of this chapter, it
persists in progressing by fives until it reaches 20, when it announces a
new base, which shows that the system will from now on be vigesimal. This
scale is one of the most interesting of which we have any record, and will
be noticed again in the next chapter. In many respects it is like the scale
of the Point Barrow Eskimo, which was given early in Chapter III. The
Eskimo languages are characteristically quinary-vigesimal in their number
systems, but few of them present such perfect examples of that method of
counting as do the two just mentioned.


  CHIPPEWAY.[298]

   1.  bejig.
   2.  nij.
   3.  nisswi.
   4.  niwin.
   5.  nanun.
   6.  ningotwasswi = 1 again?
   7.  nijwasswi    = 2 again?
   8.  nishwasswi   = 3 again?
   9.  jangasswi    = 4 again?
  10.  midasswi     = 5 again.


  MASSACHUSETTS.[299]

   1.  nequt.
   2.  neese.
   3.  nish.
   4.  yaw.
   5.  napanna     = on one side, _i.e._ 1 hand.
   6.  nequttatash = 1 added.
   7.  nesausuk    = 2 again?
   8.  shawosuk    = 3 again?
   9.  pashoogun   = it comes near, _i.e._ to 10.
  10.  puik.


  OJIBWA OF CHEGOIMEGON.[300]

   1.  bashik.
   2.  neensh.
   3.  niswe.
   4.  newin.
   5.  nanun.
   6.  ningodwaswe = 1 again?
   7.  nishwaswe   = 2 again?
   8.  shouswe     = 3 again?
   9.  shangaswe   = 4 again?
  10.  medaswe     = 5 again?


  OTTAWA.

   1.  ningotchau.
   2.  ninjwa.
   3.  niswa.
   4.  niwin.
   5.  nanau.
   6.  ningotwaswi = 1 again?
   7.  ninjwaswi   = 2 again?
   8.  nichwaswi   = 3 again?
   9.  shang.
  10.  kwetch.


  DELAWARE.

   1.  n'gutti.
   2.  niskha.
   3.  nakha.
   4.  newa.
   5.  nalan [akin to palenach, hand].
   6.  guttash  = 1 on the other side.
   7.  nishash  = 2 on the other side.
   8.  khaash   = 3 on the other side.
   9.  peshgonk = coming near.
  10.  tellen   = no more.


  SHAWNOE.

   1.  negote.
   2.  neshwa.
   3.  nithuie.
   4.  newe.
   5.  nialinwe     = gone.
   6.  negotewathwe = 1 further.
   7.  neshwathwe   = 2 further.
   8.  sashekswa    = 3 further?
   9.  chakatswe [akin to chagisse, "used up"].
  10.  metathwe     = no further.


  MICMAC.[301]

   1.  naiookt.
   2.  tahboo.
   3.  seest.
   4.  naioo.
   5.  nahn.
   6.  usoo-cum.
   7.  eloo-igunuk.
   8.  oo-gumoolchin.
   9.  pescoonaduk.
  10.  mtlin.

One peculiarity of the Micmac numerals is most noteworthy. The numerals are
real verbs, instead of adjectives, or, as is sometimes the case, nouns.
They are conjugated through all the variations of mood, tense, person, and
number. The forms given above are not those that would be used in counting,
but are for specific use, being varied according to the thought it was
intended to express. For example, _naiooktaich_ = there is 1, is present
tense; _naiooktaichcus_, there was 1, is imperfect; and _encoodaichdedou_,
there will be 1, is future. The variation in person is shown by the
following inflection:


  PRESENT TENSE.

  1st pers.  tahboosee-ek       = there are 2 of us.
  2d pers.   tahboosee-yok      = there are 2 of you.
  3d pers.   tahboo-sijik       = there are 2 of them.


  IMPERFECT TENSE.

  1st pers.  tahboosee-egup     = there were 2 of us.
  2d pers.   tahboosee-yogup    = there were 2 of you.
  3d pers.   tahboosee-sibunik  = there were 2 of them.


  FUTURE TENSE.

  3d pers.   tahboosee-dak      = there will be 2 of them, etc.

The negative form is also comprehended in the list of possible variations.
Thus, _tahboo-seekw_, there are not 2 of them; _mah tahboo-seekw_, there
will not be 2 of them; and so on, through all the changes which the
conjugation of the verb permits.


  OLD ALGONQUIN.

   1.  peygik.
   2.  ninsh.
   3.  nisswey.
   4.  neyoo.
   5.  nahran = gone.
   6.  ningootwassoo = 1 on the other side.
   7.  ninshwassoo   = 2 on the other side.
   8.  nisswasso     = 3 on the other side.
   9.  shangassoo [akin to chagisse, "used up"].
  10.  mitassoo      = no further.


  OMAHA.

   1.  meeachchee.
   2.  nomba.
   3.  rabeenee.
   4.  tooba.
   5.  satta       = hand, _i.e._ all the fingers turned down.
   6.  shappai     = 1 more.
   7.  painumba    = fingers 2.
   8.  pairabeenee = fingers 3.
   9.  shonka      = only 1 finger (remains).
  10.  kraibaira   = unbent.[302]


  CHOCTAW.

   1.  achofee.
   2.  tuklo.
   3.  tuchina.
   4.  ushta.
   5.  tahlape = the first hand ends.
   6.  hanali.
   7.  untuklo   = again 2.
   8.  untuchina = again 3.
   9.  chokali   = soon the end; _i.e._ next the last.
  10.  pokoli.


  CADDOE.

   1.  kouanigh.
   2.  behit.
   3.  daho.
   4.  hehweh.
   5.  dihsehkon.
   6.  dunkeh.
   7.  bisekah     = 5 + 2.
   8.  dousehka    = 5 + 3.
   9.  hehwehsehka = 4 + hand.
  10.  behnehaugh.


  CHIPPEWAY.

   1.  payshik.
   2.  neesh.
   3.  neeswoy.
   4.  neon.
   5.  naman = gone.
   6.  nequtwosswoy  = 1 on the other side.
   7.  neeshswosswoy = 2 on the other side.
   8.  swoswoy       = 3 on the other side?
   9.  shangosswoy [akin to chagissi, "used up"].
  10.  metosswoy     = no further.


  ADAIZE.

   1.  nancas.
   2.  nass.
   3.  colle.
   4.  tacache.
   5.  seppacan.
   6.  pacanancus = 5 + 1.
   7.  pacaness   = 5 + 2.
   8.  pacalcon   = 5 + 3.
   9.  sickinish  = hands minus?
  10.  neusne.


  PAWNEE.

   1.  askoo.
   2.  peetkoo.
   3.  touweet.
   4.  shkeetiksh.
   5.  sheeooksh    = hands half.
   6.  sheekshabish = 5 + 1.
   7.  peetkoosheeshabish = 2 + 5.
   8.  touweetshabish = 3 + 5.
   9.  looksheereewa  = 10 - 1.
  10.  looksheeree    = 2d 5?


  MINSI.

   1.  gutti.
   2.  niskha.
   3.  nakba.
   4.  newa.
   5.  nulan    = gone?
   6.  guttash  = 1 added.
   7.  nishoash = 2 added.
   8.  khaash   = 3 added.
   9.  noweli.
  10.  wimbat.


  KONLISCHEN.

   1.  tlek.
   2.  tech.
   3.  nezk.
   4.  taakun.
   5.  kejetschin.
   6.  klet uschu    = 5 + 1.
   7.  tachate uschu = 5 + 2.
   8.  nesket uschu  = 5 + 3.
   9.  kuschok       = 10 - 1?
  10.  tschinkat.


  TLINGIT.[303]

   1.  tlek.
   2.  deq.
   3.  natsk.
   4.  dak'on = 2d 2.
   5.  kedjin = hand.
   6.  tle durcu    = other 1.
   7.  daqa durcu   = other 2.
   8.  natska durcu = other 3.
   9.  gocuk.
  10.  djinkat = both hands.


  RAPID, OR FALL, INDIANS.

   1.  karci.
   2.  neece.
   3.  narce.
   4.  nean.
   5.  yautune.
   6.  neteartuce  = 1 over?
   7.  nesartuce   = 2 over?
   8.  narswartuce = 3 over?
   9.  anharbetwartuce = 4 over?
  10.  mettartuce  = no further?


  HEILTSUK.[304]

   1.  men.
   2.  matl.
   3.  yutq.
   4.  mu.
   5.  sky'a.
   6.  katla.
   7.  matlaaus = other 2?
   8.  yutquaus = other 3?
   9.  mamene   = 10 - 1.
  10.  aiky'as.


  NOOTKA.[305]

   1.  nup.
   2.  atla.
   3.  katstsa.
   4.  mo.
   5.  sutca.
   6.  nopo       = other 1?
   7.  atlpo      = other 2?
   8.  atlakutl   = 10 - 2.
   9.  ts'owakutl = 10 - 1.
  10.  haiu.


  TSIMSHIAN.[306]

   1.  gyak.
   2.  tepqat.
   3.  guant.
   4.  tqalpq.
   5.  kctonc (from _anon_, hand).
   6.  kalt     = 2d 1.
   7.  t'epqalt = 2d 2.
   8.  guandalt = 2d 3?
   9.  kctemac.
  10.  gy'ap.


  BILQULA.[306]

   1.  (s)maotl.
   2.  tlnos.
   3.  asmost.
   4.  mos.
   5.  tsech.
   6.  tqotl    = 2d 1?
   7.  nustlnos = 2d 2?
   8.  k'etlnos = 2 × 4.
   9.  k'esman.
  10.  tskchlakcht.


  MOLELE.[307]

   1.  mangu.
   2.  lapku.
   3.  mutka.
   4.  pipa.
   5.  pika.
   6.  napitka  = 1 + 5.
   7.  lapitka  = 2 + 5.
   8.  mutpitka = 3 + 5.
   9.  laginstshiatkus.
  10.  nawitspu.


  WAIILATPU.[308]

   1.  na.
   2.  leplin.
   3.  matnin.
   4.  piping.
   5.  tawit.
   6.  noina  = [5] + 1.
   7.  noilip = [5] + 2.
   8.  noimat = [5] + 3.
   9.  tanauiaishimshim.
  10.  ningitelp.


  LUTUAMI.[307]

   1.  natshik.
   2.  lapit.
   3.  ntani.
   4.  wonip.
   5.  tonapni.
   6.  nakskishuptane  = 1 + 5.
   7.  tapkishuptane   = 2 + 5.
   8.  ndanekishuptane = 3 + 5.
   9.  natskaiakish    = 10 - 1.
  10.  taunip.


  SASTE (SHASTA).[309]

   1.  tshiamu.
   2.  hoka.
   3.  hatski.
   4.  irahaia.
   5.  etsha.
   6.  tahaia.
   7.  hokaikinis  = 2 + 5.
   8.  hatsikikiri = 3 + 5.
   9.  kirihariki-ikiriu.
  10.  etsehewi.


  CAHUILLO.[310]

   1.  supli.
   2.  mewi.
   3.  mepai.
   4.  mewittsu.
   5.  nomekadnun.
   6.  kadnun-supli = 5-1.
   7.  kan-munwi    = 5-2.
   8.  kan-munpa    = 5-3.
   9.  kan-munwitsu = 5-4.
  10.  nomatsumi.


  TIMUKUA.[311]

   1.  yaha.
   2.  yutsa.
   3.  hapu.
   4.  tseketa.
   5.  marua.
   6.  mareka       = 5 + 1
   7.  pikitsa      = 5 + 2
   8.  pikinahu     = 5 + 3
   9.  peke-tsaketa = 5 + 4
  10.  tuma.


  OTOMI[312]

   1.  nara.
   2.  yocho.
   3.  chiu.
   4.  gocho.
   5.  kuto.
   6.  rato   = 1 + 5.
   7.  yoto   = 2 + 5.
   8.  chiato = 3 + 5.
   9.  guto   = 4 + 5.
  10.  reta.


  TARASCO.[313]

   1.  ma.
   2.  dziman.
   3.  tanimo.
   4.  tamu.
   5.  yumu.
   6.  kuimu.
   7.  yun-dziman = [5] + 2.
   8.  yun-tanimo = [5] + 3.
   9.  yun-tamu   = [5] + 4.
  10.  temben.


  MATLALTZINCAN.[314]

   1.  indawi.
   2.  inawi.
   3.  inyuhu.
   4.  inkunowi.
   5.  inkutaa.
   6.  inda-towi     = 1 + 5.
   7.  ine-towi      = 2 + 5.
   8.  ine-ukunowi   = 2-4.
   9.  imuratadahata = 10 - 1?
  10.  inda-hata.


  CORA.[315]

   1.  ceaut.
   2.  huapoa.
   3.  huaeica.
   4.  moacua.
   5.  anxuvi.
   6.  a-cevi    = [5] + 1.
   7.  a-huapoa  = [5] + 2.
   8.  a-huaeica = [5] + 3.
   9.  a-moacua  = [5] + 4.
  10.  tamoamata (akin to moamati, "hand").


  AYMARA.[316]

   1.  maya.
   2.  paya.
   3.  kimsa.
   4.  pusi.
   5.  piska.
   6.  tsokta.
   7.  pa-kalko    = 2 + 5.
   8.  kimsa-kalko = 3 + 5.
   9.  pusi-kalko  = 4 + 5.
  10.  tunka.


  CARIBS OF ESSEQUIBO, GUIANA.[317]

   1.  oween.
   2.  oko.
   3.  oroowa.
   4.  oko-baimema.
   5.  wineetanee     = 1 hand.
   6.  owee-puimapo   = 1 again?
   7.  oko-puimapo    = 2 again?
   8.  oroowa-puimapo = 3 again?
   9.  oko-baimema-puimapo = 4 again?
  10.  oween-abatoro.


  CARIB.[318] (ROUCOUYENNE?)

   1.  aban, amoin.
   2.  biama.
   3.  eleoua.
   4.  biam-bouri = 2 again?
   5.  ouacabo-apourcou-aban-tibateli.
   6.  aban laoyagone-ouacabo-apourcou.
   7.  biama laoyagone-ouacabo-apourcou.
   8.  eleoua laoyagone-ouacabo-apourcou.
   9.  ----
  10.  chon noucabo.

It is unfortunate that the meanings of these remarkable numerals cannot be
given. The counting is evidently quinary, but the terms used must have been
purely descriptive expressions, having their origin undoubtedly in certain
gestures or finger motions. The numerals obtained from this region, and
from the tribes to the south and east of the Carib country, are especially
rich in digital terms, and an analysis of the above numerals would probably
show clearly the mental steps through which this people passed in
constructing the rude scale which served for the expression of their ideas
of number.


  KIRIRI.[319]

   1.  biche.
   2.  watsani.
   3.  watsani dikie.
   4.  sumara oroba.
   5.  mi biche misa = 1 hand.
   6.  mirepri bu-biche misa sai.
   7.  mirepri watsani misa sai.
   8.  mirepri watsandikie misa sai.
   9.  mirepri sumara oraba sai.
  10.  mikriba misa sai = both hands.


  CAYUBABA[320]

   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.


  SAPIBOCONA[320]

   1.  karata.
   2.  mitia.
   3.  kurapa.
   4.  tsada.
   5.  maidara (from _arue_, hand).
   6.  karata-rirobo = 1 hand with.
   7.  mitia-rirobo  = 2 hand with.
   8.  kurapa-rirobo = 3 hand with.
   9.  tsada-rirobo  = 4 hand with.
  10.  bururutse     = hand hand.


  TICUNA.[321]

   1.  hueih.
   2.  tarepueh.
   3.  tomepueh.
   4.  aguemoujih
   5.  hueamepueh.
   6.  naïmehueapueh      = 5 + 1.
   7.  naïmehueatareh     = 5 + 2.
   8.  naïmehueatameapueh = 5 + 3.
   9.  gomeapueh          = 10 - 1.
  10.  gomeh.


  YANUA.[322]

   1.  tckini.
   2.  nanojui.
   3.  munua.
   4.  naïrojuino = 2d 2.
   5.  tenaja.
   6.  teki-natea    = 1 again?
   7.  nanojui-natea = 2 again?
   8.  munua-natea   = 3 again?
   9.  naïrojuino-natea = 4 again?
  10.  huijejuino    = 2 × 5?

The foregoing examples will show with considerable fulness the wide
dispersion of the quinary scale. Every part of the world contributes its
share except Europe, where the only exceptions to the universal use of the
decimal system are the half-dozen languages, which still linger on its
confines, whose number base is the vigesimal. Not only is there no living
European tongue possessing a quinary number system, but no trace of this
method of counting is found in any of the numerals of the earlier forms of
speech, which have now become obsolete. The only possible exceptions of
which I can think are the Greek [Greek: pempazein], to count by fives, and
a few kindred words which certainly do hint at a remote antiquity in which
the ancestors of the Greeks counted on their fingers, and so grouped their
units into fives. The Roman notation, the familiar I., II., III., IV.
(originally IIII.), V., VI., etc., with equal certainty suggests quinary
counting, but the Latin language contains no vestige of anything of the
kind, and the whole range of Latin literature is silent on this point,
though it contains numerous references to finger counting. It is quite
within the bounds of possibility that the prehistoric nations of Europe
possessed and used a quinary numeration. But of these races the modern
world knows nothing save the few scanty facts that can be gathered from the
stone implements which have now and then been brought to light. Their
languages have perished as utterly as have the races themselves, and
speculation concerning them is useless. Whatever their form of numeration
may have been, it has left no perceptible trace on the languages by which
they were succeeded. Even the languages of northern and central Europe
which were contemporary with the Greek and Latin of classical times have,
with the exception of the Celtic tongues of the extreme North-west, left
behind them but meagre traces for the modern student to work on. We presume
that the ancient Gauls and Goths, Huns and Scythians, and other barbarian
tribes had the same method of numeration that their descendants now have;
and it is a matter of certainty that the decimal scale was, at that time,
not used with the universality which now obtains; but wherever the decimal
was not used, the universal method was vigesimal; and that the quinary ever
had anything of a foothold in Europe is only to be guessed from its
presence to-day in almost all of the other corners of the world.

From the fact that the quinary is that one of the three natural scales with
the smallest base, it has been conjectured that all tribes possess, at some
time in their history, a quinary numeration, which at a later period merges
into either the decimal or the vigesimal, and thus disappears or forms with
one of the latter a mixed system.[323] In support of this theory it is
urged that extensive regions which now show nothing but decimal counting
were, beyond all reasonable doubt, quinary. It is well known, for example,
that the decimal system of the Malays has spread over almost the entire
Polynesian region, displacing whatever native scales it encountered. The
same phenomenon has been observed in Africa, where the Arab traders have
disseminated their own numeral system very widely, the native tribes
adopting it or modifying their own scales in such a manner that the Arab
influence is detected without difficulty.

In view of these facts, and of the extreme readiness with which a tribe
would through its finger counting fall into the use of the quinary method,
it does not at first seem improbable that the quinary was _the_ original
system. But an extended study of the methods of counting in vogue among the
uncivilized races of all parts of the world has shown that this theory is
entirely untenable. The decimal scale is no less simple in its structure
than the quinary; and the savage, as he extends the limit of his scale from
5 to 6, may call his new number 5-1, or, with equal probability, give it an
entirely new name, independent in all respects of any that have preceded
it. With the use of this new name there may be associated the conception of
"5 and 1 more"; but in such multitudes of instances the words employed show
no trace of any such meaning, that it is impossible for any one to draw,
with any degree of safety, the inference that the signification was
originally there, but that the changes of time had wrought changes in
verbal form so great as to bury it past the power of recovery. A full
discussion of this question need not be entered upon here. But it will be
of interest to notice two or three numeral scales in which the quinary
influence is so faint as to be hardly discernible. They are found in
considerable numbers among the North American Indian languages, as may be
seen by consulting the vocabularies that have been prepared and published
during the last half century.[324] From these I have selected the
following, which are sufficient to illustrate the point in question:


  QUAPPA.

   1.  milchtih.
   2.  nonnepah.
   3.  dahghenih.
   4.  tuah.
   5.  sattou.
   6.  schappeh.
   7.  pennapah.
   8.  pehdaghenih.
   9.  schunkkah.
  10.  gedeh bonah.


  TERRABA.[325]

   1.  krara.
   2.  krowü.
   3.  krom miah.
   4.  krob king.
   5.  krasch kingde.
   6.  terdeh.
   7.  kogodeh.
   8.  kwongdeh.
   9.  schkawdeh.
  10.  dwowdeh.


  MOHICAN

   1.  ngwitloh.
   2.  neesoh.
   3.  noghhoh.
   4.  nauwoh.
   5.  nunon.
   6.  ngwittus.
   7.  tupouwus.
   8.  ghusooh.
   9.  nauneeweh.
  10.  mtannit.

In the Quappa scale 7 and 8 appear to be derived from 2 and 3, while 6 and
9 show no visible trace of kinship with 1 and 4. In Mohican, on the other
hand, 6 and 9 seem to be derived from 1 and 4, while 7 and 8 have little or
no claim to relationship with 2 and 3. In some scales a single word only is
found in the second quinate to indicate that 5 was originally the base on
which the system rested. It is hardly to be doubted, even, that change
might affect each and every one of the numerals from 5 to 10 or 6 to 9, so
that a dependence which might once have been easily detected is now
unrecognizable.

But if this is so, the natural and inevitable question follows--might not
this have been the history of all numeral scales now purely decimal? May
not the changes of time have altered the compounds which were once a clear
indication of quinary counting, until no trace remains by which they can be
followed back to their true origin? Perhaps so. It is not in the least
degree probable, but its possibility may, of course, be admitted. But even
then the universality of quinary counting for primitive peoples is by no
means established. In Chapter II, examples were given of races which had no
number base. Later on it was observed that in Australia and South America
many tribes used 2 as their number base; in some cases counting on past 5
without showing any tendency to use that as a new unit. Again, through the
habit of counting upon the finger joints, instead of the fingers
themselves, the use of 3 as a base is brought into prominence, and 6 and 9
become 2 threes and 3 threes, respectively, instead of 5 + 1 and 5 + 4. The
same may be noticed of 4. Counting by means of his fingers, without
including the thumbs, the savage begins by dividing into fours instead of
fives. Traces of this form of counting are somewhat numerous, especially
among the North American aboriginal tribes. Hence the quinary form of
counting, however widespread its use may be shown to be, can in no way be
claimed as the universal method of any stage of development in the history
of mankind.

In the vast majority of cases, the passage from the base to the next
succeeding number in any scale, is clearly defined. But among races whose
intelligence is of a low order, or--if it be permissible to express it in
this way--among races whose number sense is feeble, progression from one
number to the next is not always in accordance with any well-defined law.
After one or two distinct numerals the count may, as in the case of the
Veddas and the Andamans, proceed by finger pantomime and by the repetition
of the same word. Occasionally the same word is used for two successive
numbers, some gesture undoubtedly serving to distinguish the one from the
other in the savage's mind. Examples of this are not infrequent among the
forest tribes of South America. In the Tariana dialect 9 and 10 are
expressed by the same word, _paihipawalianuda;_ in Cobeu, 8 and 9 by
_pepelicoloblicouilini;_ in Barre, 4, 5, and 9 by _ualibucubi._[326] In
other languages the change from one numeral to the next is so slight that
one instinctively concludes that the savage is forming in his own mind
another, to him new, numeral immediately from the last. In such cases the
entire number system is scanty, and the creeping hesitancy with which
progress is made is visible in the forms which the numerals are made to
take. A single illustration or two of this must suffice; but the ones
chosen are not isolated cases. The scale of the Macunis,[327] one of the
numerous tribes of Brazil, is

   1.  pocchaenang.
   2.  haihg.
   3.  haigunhgnill.
   4.  haihgtschating.
   5.  haihgtschihating     = another 4?
   6.  hathig-stchihathing  = 2-4?
   7.  hathink-tschihathing = 2-5?
   8.  hathink-tschihating  = 2 × 4?

The complete absence of--one is tempted to say--any rhyme or reason from
this scale is more than enough to refute any argument which might tend to
show that the quinary, or any other scale, was ever the sole number scale
of primitive man. Irregular as this is, the system of the Montagnais fully
matches it, as the subjoined numerals show:[328]

   1.  inl'are.
   2.  nak'e.
   3.  t'are.
   4.  dinri.
   5.  se-sunlare.
   6.  elkke-t'are      = 2 × 3.
   7.  t'a-ye-oyertan   = 10 - 3,
        or inl'as dinri = 4 + 3?
   8.  elkke-dinri      = 2 × 4.
   9.  inl'a-ye-oyertan = 10 - 1.
  10.  onernan.





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 numeration are
found indifferently in warm and in cold countries; in fruitful and in
barren lands; in maritime and in inland regions; and among highly civilized
or deeply degraded peoples.

Whether or not the principal number base of any tribe is to be 20 seems to
depend entirely upon a single consideration; are the fingers alone used as
an aid to counting, or are both fingers and toes used? If only the fingers
are employed, the resulting scale must become decimal if sufficiently
extended. If use is made of the toes in addition to the fingers, the
outcome must inevitably be a vigesimal system. Subordinate to either one of
these the quinary may and often does appear. It is never the principal base
in any extended system.

To the statement just made respecting the origin of vigesimal counting,
exception may, of course, be taken. In the case of numeral scales like the
Welsh, the Nahuatl, and many others where the exact meanings of the
numerals cannot be ascertained, no proof exists that the ancestors of these
peoples ever used either finger or toe counting; and the sweeping statement
that any vigesimal scale is the outgrowth of the use of these natural
counters is not susceptible of proof. But so many examples are met with in
which the origin is clearly of this nature, that no hesitation is felt in
putting the above forward as a general explanation for the existence of
this kind of counting. Any other origin is difficult to reconcile with
observed facts, and still more difficult to reconcile with any rational
theory of number system development. Dismissing from consideration the
quinary scale, let us briefly examine once more the natural process of
evolution through which the decimal and the vigesimal scales come into
being. After the completion of one count of the fingers the savage
announces his result in some form which definitely states to his mind the
fact that the end of a well-marked series has been reached. Beginning
again, he now repeats his count of 10, either on his own fingers or on the
fingers of another. With the completion of the second 10 the result is
announced, not in a new unit, but by means of a duplication of the term
already used. It is scarcely credible that the unit unconsciously adopted
at the termination of the first count should now be dropped, and a new one
substituted in its place. When the method here described is employed, 20 is
not a natural unit to which higher numbers may be referred. It is wholly
artificial; and it would be most surprising if it were adopted. But if the
count of the second 10 is made on the toes in place of the fingers, the
element of repetition which entered into the previous method is now
wanting. Instead of referring each new number to the 10 already completed,
the savage is still feeling his way along, designating his new terms by
such phrases as "1 on the foot," "2 on the other foot," etc. And now, when
20 is reached, a single series is finished instead of a double series as
before; and the result is expressed in one of the many methods already
noticed--"one man," "hands and feet," "the feet finished," "all the fingers
of hands and feet," or some equivalent formula. Ten is no longer the
natural base. The number from which the new start is made is 20, and the
resulting scale is inevitably vigesimal. If pebbles or sticks are used
instead of fingers, the system will probably be decimal. But back of the
stick and pebble counting the 10 natural counters always exist, and to them
we must always look for the origin of this scale.

In any collection of the principal vigesimal number systems of the world,
one would naturally begin with those possessed by the Celtic races of
Europe. These races, the earliest European peoples of whom we have any
exact knowledge, show a preference for counting by twenties, which is
almost as decided as that manifested by Teutonic races for counting by
tens. It has been conjectured by some writers that the explanation for this
was to be found in the ancient commercial intercourse which existed between
the Britons and the Carthaginians and Phoenicians, whose number systems
showed traces of a vigesimal tendency. Considering the fact that the use of
vigesimal counting was universal among Celtic races, this explanation is
quite gratuitous. The reason why the Celts used this method is entirely
unknown, and need not concern investigators in the least. But the fact that
they did use it is important, and commands attention. The five Celtic
languages, Breton, Irish, Welsh, Manx, and Gaelic, contain the following
well-defined vigesimal scales. Only the principal or characteristic
numerals are given, those being sufficient to enable the reader to follow
intelligently the growth of the systems. Each contains the decimal element
also, and is, therefore, to be regarded as a mixed decimal-vigesimal
system.


  IRISH.[329]

    10.  deic.
    20.  fice.
    30.  triocad      = 3-10
    40.  da ficid     = 2-20.
    50.  caogad       = 5-10.
    60.  tri ficid    = 3-20.
    70.  reactmoga    = 7-10.
    80.  ceitqe ficid = 4-20.
    90.  nocad        = 9-10.
   100.  cead.
  1000.  mile.


  GAELIC.[330]

    10.  deich.
    20.  fichead.
    30.  deich ar fichead         = 10 + 20.
    40.  da fhichead              = 2-20.
    50.  da fhichead is deich     = 40 + 10.
    60.  tri fichead              = 3-20.
    70.  tri fichead is deich     = 60 + 10.
    80.  ceithir fichead          = 4-20.
    90.  ceithir fichead is deich = 80 + 10.
   100.  ceud.
  1000.  mile.


  WELSH.[331]

    10.  deg.
    20.  ugain.
    30.  deg ar hugain      = 10 + 20.
    40.  deugain            = 2-20.
    50.  deg a deugain      = 10 + 40.
    60.  trigain            = 3-20.
    70.  deg a thrigain     = 10 + 60.
    80.  pedwar ugain       = 4-20.
    90.  deg a pedwar ugain = 80 + 10.
   100.  cant.


  MANX.[332]

    10.  jeih.
    20.  feed.
    30.  yn jeih as feed    = 10 + 20.
    40.  daeed              = 2-20.
    50.  jeih as daeed      = 10 + 40.
    60.  three-feed         = 3-20.
    70.  three-feed as jeih = 60 + 10.
    80.  kiare-feed         = 4-20.
   100.  keead.
  1000.  thousane, or jeih cheead.


  BRETON.[333]

    10.  dec.
    20.  ueguend.
    30.  tregond             = 3-10.
    40.  deu ueguend         = 2-20.
    50.  hanter hand         = half hundred.
    60.  tri ueguend         = 3-20.
    70.  dec ha tri ueguend  = 10 + 60.
    80.  piar ueguend        = 4-20.
    90.  dec ha piar ueguend = 10 + 80.
   100.  cand.
   120.  hueh ueguend    = 6-20.
   140.  seih ueguend    = 7-20.
   160.  eih ueguend     = 8-20.
   180.  nau ueguend     = 9-20.
   200.  deu gand        = 2-100.
   240.  deuzec ueguend  = 12-20.
   280.  piarzec ueguend = 14-20.
   300.  tri hand, or pembzec ueguend.
   400.  piar hand       = 4-100.
  1000.  mil.

These lists show that the native development of the Celtic number systems,
originally showing a strong preference for the vigesimal method of
progression, has been greatly modified by intercourse with Teutonic and
Latin races. The higher numerals in all these languages, and in Irish many
of the lower also, are seen at a glance to be decimal. Among the scales
here given the Breton, the legitimate descendant of the ancient Gallic, is
especially interesting; but here, just as in the other Celtic tongues, when
we reach 1000, the familiar Latin term for that number appears in the
various corruptions of _mille_, 1000, which was carried into the Celtic
countries by missionary and military influences.

In connection with the Celtic language, mention must be made of the
persistent vigesimal element which has held its place in French. The
ancient Gauls, while adopting the language of their conquerors, so far
modified the decimal system of Latin as to replace the natural _septante_,
70, _octante_, 80, _nonante_, 90, by _soixante-dix_, 60-10, _quatre-vingt_,
4-20, and _quatrevingt-dix_, 4-20-10. From 61 to 99 the French method of
counting is wholly vigesimal, except for the presence of the one word
_soixante_. In old French this element was still more pronounced.
_Soixante_ had not yet appeared; and 60 and 70 were _treis vinz_, 3-20, and
_treis vinz et dis_, 3-20 and 10 respectively. Also, 120 was _six vinz_,
6-20, 140 was _sept-vinz_, etc.[334] How far this method ever extended in
the French language proper, it is, perhaps, impossible to say; but from the
name of an almshouse, _les quinze-vingts_,[335] which formerly existed in
Paris, and was designed as a home for 300 blind persons, and from the
_pembzek-ueguent_, 15-20, of the Breton, which still survives, we may infer
that it was far enough to make it the current system of common life.

Europe yields one other example of vigesimal counting, in the number system
of the Basques. Like most of the Celtic scales, the Basque seems to become
decimal above 100. It does not appear to be related to any other European
system, but to be quite isolated philologically. The higher units, as
_mila_, 1000, are probably borrowed, and not native. The tens in the Basque
scale are:[336]

    10.  hamar.
    20.  hogei.
    30.  hogei eta hamar     = 20 + 10.
    40.  berrogei            = 2-20.
    50.  berrogei eta hamar  = 2-20 + 10.
    60.  hirurogei           = 3-20.
    70.  hirurogei eta hamar = 3-20 + 10.
    80.  laurogei            = 4-20.
    90.  laurogei eta hamar  = 4-20 + 10.
   100.  ehun.
  1000.  _milla_.

Besides these we find two or three numeral scales in Europe which contain
distinct traces of vigesimal counting, though the scales are, as a whole,
decidedly decimal. The Danish, one of the essentially Germanic languages,
contains the following numerals:

    30.  tredive            = 3-10.
    40.  fyrretyve          = 4-10.
    50.  halvtredsindstyve  = half (of 20) from 3-20.
    60.  tresindstyve       = 3-20.
    70.  halvfierdsindstyve = half from 4-20.
    80.  fiirsindstyve      = 4-20.
    90.  halvfemsindstyve   = half from 5-20.
   100.  hundrede.

Germanic number systems are, as a rule, pure decimal systems; and the
Danish exception is quite remarkable. We have, to be sure, such expressions
in English as _three score_, _four score_, etc., and the Swedish,
Icelandic, and other languages of this group have similar terms. Still,
these are not pure numerals, but auxiliary words rather, which belong to
the same category as _pair_, _dozen_, _dizaine_, etc., while the Danish
words just given are the ordinary numerals which form a part of the
every-day vocabulary of that language. The method by which this scale
expresses 50, 70, and 90 is especially noticeable. It will be met with
again, and further examples of its occurrence given.

In Albania there exists one single fragment of vigesimal numeration, which
is probably an accidental compound rather than the remnant of a former
vigesimal number system. With this single exception the Albanian scale is
of regular decimal formation. A few of the numerals are given for the sake
of comparison:[337]

    30.  tridgiete    = 3-10.
    40.  dizet        = 2-20.
    50.  pesedgiete   = 5-10.
    60.  giastedgiete = 6-10, etc.

Among the almost countless dialects of Africa we find a comparatively small
number of vigesimal number systems. The powers of the negro tribes are not
strongly developed in counting, and wherever their numeral scales have been
taken down by explorers they have almost always been found to be decimal or
quinary-decimal. The small number I have been able to collect are here
given. They are somewhat fragmentary, but are as complete as it was
possible to make them.


  AFFADEH.[338]

    10.  dekang.
    20.  degumm.
    30.  piaske.
    40.  tikkumgassih      = 20 × 2.
    50.  tikkumgassigokang = 20 × 2 + 10.
    60.  tikkumgakro       = 20 × 3.
    70.  dungokrogokang    = 20 × 3 + 10.
    80.  dukumgade         = 20 × 4.
    90.  dukumgadegokang   = 20 × 4 + 10.
   100.  miah (borrowed from the Arabs).


  IBO.[339]

    10.  iri.
    20.  ogu.
    30.  ogu n-iri    = 20 + 10,
          or iri ato  = 10 × 3.
    40.  ogu abuo     = 20 × 2,
          or iri anno = 10 × 4.
   100.  ogu ise      = 20 × 5.


  VEI.[340]

    10.  tan.
    20.  mo bande         = a person finished.
    30.  mo bande ako tan = 20 + 10.
    40.  mo fera bande    = 2 × 20.
   100.  mo soru bande    = 5 persons finished.


  YORUBA.[341]

    10.  duup.
    20.  ogu.
    30.  ogbo.
    40.  ogo-dzi = 20 × 2.
    60.  ogo-ta  = 20 × 3.
    80.  ogo-ri  = 20 × 4.
   100.  ogo-ru  = 20 × 5.
   120.  ogo-fa  = 20 × 6.
   140.  ogo-dze = 20 × 7.
   160.  ogo-dzo = 20 × 8, etc.


  EFIK.[342]

    10.  duup.
    20.  edip.
    30.  edip-ye-duup = 20 + 10.
    40.  aba  = 20 × 2.
    60.  ata  = 20 × 3.
    80.  anan = 20 × 4.
   100.  ikie.

The Yoruba scale, to which reference has already been made, p. 70, again
shows its peculiar structure, by continuing its vigesimal formation past
100 with no interruption in its method of numeral building. It will be
remembered that none of the European scales showed this persistency, but
passed at that point into decimal numeration. This will often be found to
be the case; but now and then a scale will come to our notice whose
vigesimal structure is continued, without any break, on into the hundreds
and sometimes into the thousands.


  BONGO.[343]

    10.  kih.
    20.  mbaba kotu  = 20 × 1.
    40.  mbaba gnorr = 20 × 2.
   100.  mbaba mui   = 20 × 5.


  MENDE.[344]

    10.  pu.
    20.  nu yela gboyongo mai = a man finished.
    30.  nu yela gboyongo mahu pu = 20 + 10.
    40.  nu fele gboyongo = 2 men finished.
   100.  nu lolu gboyongo = 5 men finished.


  NUPE.[345]

    10.  gu-wo.
    20.  esin.
    30.  gbonwo.
    40.  si-ba = 2 × 20.
    50.  arota.
    60.  sita  = 3 × 20.
    70.  adoni.
    80.  sini  = 4 × 20.
    90.  sini be-guwo = 80 + 10.
   100.  sisun = 5 × 20.


  LOGONE.[346]

    10.  chkan.
    20.  tkam.
    30.  tkam ka chkan      = 20 + 10.
    40.  tkam ksde          = 20 × 2.
    50.  tkam ksde ka chkan = 40 + 10.
    60.  tkam gachkir       = 20 × 3.
   100.  mia (from Arabic).
  1000.  debu.


  MUNDO.[347]

    10.  nujorquoi.
    20.  tiki bere.
    30.  tiki bire nujorquoi  = 20 + 10.
    40.  tiki borsa           = 20 × 2.
    50.  tike borsa nujorquoi = 40 + 10.


  MANDINGO.[348]

    10.  tang.
    20.  mulu.
    30.  mulu nintang       = 20 + 10.
    40.  mulu foola         = 20 × 2.
    50.  mulu foola nintang = 40 + 10.
    60.  mulu sabba         = 20 × 3.
    70.  mulu sabba nintang = 60 + 10.
    80.  mulu nani          = 20 × 4.
    90.  mulu nani nintang  = 80 + 10.
   100.  kemi.

This completes the scanty list of African vigesimal number systems that a
patient and somewhat extended search has yielded. It is remarkable that the
number is no greater. Quinary counting is not uncommon in the "Dark
Continent," and there is no apparent reason why vigesimal reckoning should
be any less common than quinary. Any one investigating African modes of
counting with the material at present accessible, will find himself
hampered by the fact that few explorers have collected any except the first
ten numerals. This leaves the formation of higher terms entirely unknown,
and shows nothing beyond the quinary or non-quinary character of the
system. Still, among those which Stanley, Schweinfurth, Salt, and others
have collected, by far the greatest number are decimal. As our knowledge of
African languages is extended, new examples of the vigesimal method may be
brought to light. But our present information leads us to believe that they
will be few in number.

In Asia the vigesimal system is to be found with greater frequency than in
Europe or Africa, but it is still the exception. As Asiatic languages are
much better known than African, it is probable that the future will add but
little to our stock of knowledge on this point. New instances of counting
by twenties may still be found in northern Siberia, where much ethnological
work yet remains to be done, and where a tendency toward this form of
numeration has been observed to exist. But the total number of Asiatic
vigesimal scales must always remain small--quite insignificant in
comparison with those of decimal formation.

In the Caucasus region a group of languages is found, in which all but
three or four contain vigesimal systems. These systems are as follows:


  ABKHASIA.[349]

    10.  zpha-ba.
    20.  gphozpha         = 2 × 10.
    30.  gphozphei zphaba = 20 + 10.
    40.  gphin-gphozpha   = 2 × 20.
    60.  chin-gphozpha    = 3 × 20.
    80.  phsin-gphozpha   = 4 × 20.
   100.  sphki.


  AVARI

    10.  antsh-go.
    20.  qo-go.
    30.  lebergo.
    40.  khi-qogo           = 2 × 20.
    50.  khiqojalda antshgo = 40 + 10.
    60.  lab-qogo           = 3 × 20.
    70.  labqojalda antshgo = 60 + 10.
    80.  un-qogo            = 4 × 20.
   100.  nusgo.


  KURI

    10.  tshud.
    20.  chad.
    30.  channi tshud = 20 + 10.
    40.  jachtshur.
    50.  jachtshurni tshud = 40 + 10.
    60.  put chad          = 3 × 20.
    70.  putchanni tshud   = 60 + 10.
    80.  kud-chad          = 4 × 20.
    90.  kudchanni tshud   = 80 + 10.
   100.  wis.


  UDI

    10.  witsh.
    20.  qa.
    30.  sa-qo-witsh   = 20 + 10.
    40.  pha-qo        = 2 × 20.
    50.  pha-qo-witsh  = 40 + 10.
    60.  chib-qo       = 3 × 20.
    70.  chib-qo-witsh = 60 + 10.
    80.  bip-qo        = 4 × 20.
    90.  bip-qo-witsh  = 80 + 10.
   100.  bats.
  1000.  hazar (Persian).


  TCHETCHNIA

    10.  ith.
    20.  tqa.
    30.  tqe ith       = 20 + 10.
    40.  sauz-tqa      = 2 × 20.
    50.  sauz-tqe ith  = 40 + 10.
    60.  chuz-tqa      = 3 × 20.
    70.  chuz-tqe ith  = 60 + 10.
    80.  w-iez-tqa     = 4 × 20.
    90.  w-iez-tqe ith = 80 + 10.
   100.  b'e.
  1000.  ezir (akin to Persian).


  THUSCH

    10.  itt.
    20.  tqa.
    30.  tqa-itt        = 20 + 10.
    40.  sauz-tq        = 2 × 20.
    50.  sauz-tqa-itt   = 40 + 10.
    60.  chouz-tq       = 3 × 20.
    70.  chouz-tqa-itt  = 60 + 10.
    80.  dhewuz-tq      = 4 × 20.
    90.  dhewuz-tqa-itt = 80 + 10.
   100.  phchauz-tq     = 5 × 20.
   200.  itsha-tq       = 10 × 20.
   300.  phehiitsha-tq  = 15 × 20.
  1000.  satsh tqauz-tqa itshatqa = 2 × 20 × 20 + 200.


  GEORGIA

    10.  athi.
    20.  otsi.
    30.  ots da athi     = 20 + 10.
    40.  or-m-otsi       = 2 × 20.
    50.  ormots da athi  = 40 + 10.
    60.  sam-otsi        = 3 × 20.
    70.  samots da athi  = 60 + 10.
    80.  othch-m-otsi    = 4 × 20.
    90.  othmots da athi = 80 + 10.
   100.  asi.
  1000.  ath-asi = 10 × 100.


  LAZI

    10.  wit.
    20.  öts.
    30.  öts do wit         = 20 × 10.
    40.  dzur en öts        = 2 × 20.
    50.  dzur en öts do wit = 40 + 10.
    60.  dzum en öts        = 3 × 20.
    70.  dzum en öts do wit = 60 + 10.
    80.  otch-an-öts        = 4 × 20.
   100.  os.
  1000.  silia (akin to Greek).


  CHUNSAG.[350]

    10.  ants-go.
    20.  chogo.
    30.  chogela antsgo      = 20 + 10.
    40.  kichogo             = 2 × 20.
    50.  kichelda antsgo     = 40 + 10.
    60.  taw chago           = 3 × 20.
    70.  taw chogelda antsgo = 60 + 10.
    80.  uch' chogo          = 4 × 20.
    90.  uch' chogelda antsgo.
   100.  nusgo.
  1000.  asargo (akin to Persian).


  DIDO.[351]

    10.  zino.
    20.  ku.
    30.  kunozino.
    40.  kaeno ku        = 2 × 20.
    50.  kaeno kuno zino = 40 + 10.
    60.  sonno ku        = 3 × 20.
    70.  sonno kuno zino = 60 + 10.
    80.  uino ku         = 4 × 20.
    90.  uino huno zino  = 80 + 10.
   100.  bischon.
   400.  kaeno kuno zino = 40 × 10.


  AKARI

    10.  entzelgu.
    20.  kobbeggu.
    30.  lowergu.
    40.  kokawu      = 2 × 20.
    50.  kikaldanske = 40 + 10.
    60.  secikagu.
    70.  kawalkaldansku = 3 × 20 + 10.
    80.  onkuku         = 4 × 20.
    90.  onkordansku    = 4 × 20 + 10.
   100.  nosku.
  1000.  askergu (from Persian).


  CIRCASSIA

    10.  psche.
    20.  to-tsch.
    30.  totsch-era-pschirre   = 20 + 10.
    40.  ptl'i-sch             = 4 × 10.
    50.  ptl'isch-era-pschirre = 40 + 10.
    60.  chi-tsch              = 6 × 10.
    70.  chitsch-era-pschirre  = 60 + 10.
    80.  toshitl               = 20 × 4?
    90.  toshitl-era-pschirre  = 80 + 10.
   100.  scheh.
  1000.  min (Tartar) or schi-psche = 100 × 10.

The last of these scales is an unusual combination of decimal and
vigesimal. In the even tens it is quite regularly decimal, unless 80 is of
the structure suggested above. On the other hand, the odd tens are formed
in the ordinary vigesimal manner. The reason for this anomaly is not
obvious. I know of no other number system that presents the same
peculiarity, and cannot give any hypothesis which will satisfactorily
account for its presence here. In nearly all the examples given the decimal
becomes the leading element in the formation of all units above 100, just
as was the case in the Celtic scales already noticed.

Among the northern tribes of Siberia the numeral scales appear to be ruder
and less simple than those just examined, and the counting to be more
consistently vigesimal than in any scale we have thus far met with. The two
following examples are exceedingly interesting, as being among the best
illustrations of counting by twenties that are to be found anywhere in the
Old World.


  TSCHUKSCHI.[352]

    10.  migitken  = both hands.
    20.  chlik-kin = a whole man.
    30.  chlikkin mingitkin parol = 20 + 10.
    40.  nirach chlikkin = 2 × 20.
   100.  milin chlikkin  = 5 × 20.
   200.  mingit chlikkin = 10 × 20, _i.e._ 10 men.
  1000.  miligen chlin-chlikkin = 5 × 200, _i.e._ five (times) 10 men.


  AINO.[353]

    10.  wambi.
    20.  choz.
    30.  wambi i-doehoz        = 10 from 40.
    40.  tochoz                = 2 × 20.
    50.  wambi i-richoz        = 10 from 60.
    60.  rechoz                = 3 × 20.
    70.  wambi [i?] inichoz    = 10 from 80.
    80.  inichoz               = 4 × 20.
    90.  wambi aschikinichoz   = 10 from 100.
   100.  aschikinichoz         = 5 × 20.
   110.  wambi juwanochoz      = 10 from 120.
   120.  juwano choz           = 6 × 20.
   130.  wambi aruwanochoz     = 10 from 140.
   140.  aruwano choz          = 7 × 20.
   150.  wambi tubischano choz = 10 from 160.
   160.  tubischano choz       = 8 × 20.
   170.  wambi schnebischano choz = 10 from 180.
   180.  schnebischano choz    = 9 × 20.
   190.  wambi schnewano choz  = 10 from 200.
   200.  schnewano choz        = 10 × 20.
   300.  aschikinichoz i gaschima chnewano choz = 5 × 20 + 10 × 20.
   400.  toschnewano choz      = 2 × (10 × 20).
   500.  aschikinichoz i gaschima toschnewano choz = 100 + 400.
   600.  reschiniwano choz     = 3 × 200.
   700.  aschikinichoz i gaschima reschiniwano choz = 100 + 600.
   800.  inischiniwano choz    = 4 × 200.
   900.  aschikinichoz i gaschima inischiniwano choz = 100 + 800.
  1000.  aschikini schinewano choz = 5 × 200.
  2000.  wanu schinewano choz   = 10 × (10 × 20).

This scale is in one sense wholly vigesimal, and in another way it is not
to be regarded as pure, but as mixed. Below 20 it is quinary, and, however
far it might be extended, this quinary element would remain, making the
scale quinary-vigesimal. But in another sense, also, the Aino system is not
pure. In any unmixed vigesimal scale the word for 400 must be a simple
word, and that number must be taken as the vigesimal unit corresponding to
100 in the decimal scale. But the Ainos have no simple numeral word for any
number above 20, forming all higher numbers by combinations through one or
more of the processes of addition, subtraction, and multiplication. The
only number above 20 which is used as a unit is 200, which is expressed
merely as 10 twenties. Any even number of hundreds, or any number of
thousands, is then indicated as being so many times 10 twenties; and the
odd hundreds are so many times 10 twenties, plus 5 twenties more. This
scale is an excellent example of the cumbersome methods used by uncivilized
races in extending their number systems beyond the ordinary needs of daily
life.

In Central Asia a single vigesimal scale comes to light in the following
fragment of the Leptscha scale, of the Himalaya region:[354]

    10.  kati.
    40.  kafali      = 4 × 10,
          or kha nat = 2 × 20.
    50.  kafano      = 5 × 10,
          or kha nat sa kati = 2 × 20 + 10.
   100.  gjo, or kat.

Further to the south, among the Dravidian races, the vigesimal element is
also found. The following will suffice to illustrate the number systems of
these dialects, which, as far as the material at hand shows, are different
from each other only in minor particulars:


  MUNDARI.[355]

    10.  gelea.
    20.  mi hisi.
    30.  mi hisi gelea = 20 + 10.
    40.  bar hisi  = 2 × 20.
    60.  api hisi  = 3 × 20.
    80.  upun hisi = 4 × 20.
   100.  mone hisi = 5 × 20.

In the Nicobar Islands of the Indian Ocean a well-developed example of
vigesimal numeration is found. The inhabitants of these islands are so low
in the scale of civilization that a definite numeral system of any kind is
a source of some surprise. Their neighbours, the Andaman Islanders, it will
be remembered, have but two numerals at their command; their intelligence
does not seem in any way inferior to that of the Nicobar tribes, and one is
at a loss to account for the superior development of the number sense in
the case of the latter. The intercourse of the coast tribes with traders
might furnish an explanation of the difficulty were it not for the fact
that the numeration of the inland tribes is quite as well developed as that
of the coast tribes; and as the former never come in contact with traders
and never engage in barter of any kind except in the most limited way, the
conclusion seems inevitable that this is merely one of the phenomena of
mental development among savage races for which we have at present no
adequate explanation. The principal numerals of the inland and of the coast
tribes are:[356]


  INLAND TRIBES                        COAST TRIBES

    10.  teya.                           10.  sham.
    20.  heng-inai.                      20.  heang-inai.
    30.  heng-inai-tain                  30.  heang-inai-tanai
         = 20 + 5 (couples).                  = 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                 300.  heang-tanai-inai
         = (10 + 5) × 20.                     = (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. In the eastern portions of North America and in the extreme
western portions of South America the decimal or the quinary decimal scale
is in general use. But in the northern regions of North America, in western
Canada and northwestern United States, in Mexico and Central America, and
in the northern and western parts of South America, the unit of counting
among the great majority of the native races was 20. The ethnological
affinities of these races are not yet definitely ascertained; and it is no
part of the scope of this work to enter into any discussion of that
involved question. But either through contact or affinity, this form of
numeration spread in prehistoric times over half or more than half of the
western hemisphere. It was the method employed by the rude Eskimos of the
north and their equally rude kinsmen of Paraguay and eastern Brazil; by the
forest Indians of Oregon and British Columbia, and by their more southern
kinsmen, the wild tribes of the Rio Grande and of the Orinoco. And, most
striking and interesting of all, it was the method upon which were based
the numeral systems of the highly civilized races of Mexico, Yucatan, and
New Granada. Some of the systems obtained from the languages of these
peoples are perfect, extended examples of vigesimal counting, not to be
duplicated in any other quarter of the globe. The ordinary unit was, as
would be expected, "one man," and in numerous languages the words for 20
and man are identical. But in other cases the original meaning of that
numeral word has been lost; and in others still it has a signification
quite remote from that given above. These meanings will be noticed in
connection with the scales themselves, which are given, roughly speaking,
in their geographical order, beginning with the Eskimo of the far north.
The systems of some of the tribes are as follows:


  ALASKAN ESKIMOS.[357]

    10.  koleet.
    20.  enuenok.
    30.  enuenok kolinik   = 20 + 10.
    40.  malho kepe ak     = 2 × 20.
    50.  malho-kepe ak-kolmik che pah ak to = 2 × 20 + 10.
    60.  pingi shu-kepe ak = 3 × 20.
   100.  tale ma-kepe ak   = 5 × 20.
   400.  enue nok ke pe ak = 20 × 20.


  TCHIGLIT.[358]

    10.  krolit.
    20.  kroleti, or innun = man.
    30.  innok krolinik-tchikpalik = man + 2 hands.
    40.  innum mallerok   = 2 men.
    50.  adjigaynarmitoat = as many times 10 as the fingers of the hand.
    60.  innumipit        = 3 men.
    70.  innunmalloeronik arveneloerit = 7 men?
    80.  innun pinatçunik arveneloerit = 8 men?
    90.  innun tcitamanik arveneloerit = 9 men?
   100.  itchangnerkr.
  1000.  itchangner-park = great 100.

The meanings for 70, 80, 90, are not given by Father Petitot, but are of
such a form that the significations seem to be what are given above. Only a
full acquaintance with the Tchiglit language would justify one in giving
definite meanings to these words, or in asserting that an error had been
made in the numerals. But it is so remarkable and anomalous to find the
decimal and vigesimal scales mingled in this manner that one involuntarily
suspects either incompleteness of form, or an actual mistake.


  TLINGIT.[359]

    10.  djinkat = both hands?
    20.  tle ka  = 1 man.
    30.  natsk djinkat  = 3 × 10.
    40.  dak'on djinkat = 4 × 10.
    50.  kedjin djinkat = 5 × 10.
    60.  tle durcu djinkat    = 6 × 10.
    70.  daqa durcu djinkat   = 7 × 10.
    80.  natska durcu djinkat = 8 × 10.
    90.  gocuk durcu djinkat  = 9 × 10.
   100.  kedjin ka  = 5 men, or 5 × 20.
   200.  djinkat ka = 10 × 20.
   300.  natsk djinkat ka  = 30 men.
   400.  dak'on djinkat ka = 40 men.

This scale contains a strange commingling of decimal and vigesimal
counting. The words for 20, 100, and 200 are clear evidence of vigesimal,
while 30 to 90, and the remaining hundreds, are equally unmistakable proof
of decimal, numeration. The word _ka_, man, seems to mean either 10 or 20;
a most unusual occurrence. The fact that a number system is partly decimal
and partly vigesimal is found to be of such frequent occurrence that this
point in the Tlingit scale need excite no special wonder. But it is
remarkable that the same word should enter into numeral composition under
such different meanings.


  NOOTKA.[360]

    10.  haiu.
    20.  tsakeits.
    30.  tsakeits ic haiu = 20 + 10.
    40.  atlek    = 2 × 20.
    60.  katstsek = 3 × 20.
    80.  moyek    = 4 × 20.
   100.  sutc'ek  = 5 × 20.
   120.  nop'ok   = 6 × 20.
   140.  atlpok   = 7 × 20.
   160.  atlakutlek = 8 × 20.
   180.  ts'owakutlek = 9 × 20.
   200.  haiuk    = 10 × 20.

This scale is quinary-vigesimal, with no apparent decimal element in its
composition. But the derivation of some of the terms used is detected with
difficulty. In the following scale the vigesimal structure is still more
obscure.


  TSIMSHIAN.[361]

    10.  gy'ap.
    20.  kyedeel = 1 man.
    30.  gulewulgy'ap.
    40.  t'epqadalgyitk, or tqalpqwulgyap.
    50.  kctoncwulgyap.
   100.  kcenecal.
   200.  k'pal.
   300.  k'pal te kcenecal = 200 + 100.
   400.  kyedal.
   500.  kyedal te kcenecal = 400 + 100.
   600.  gulalegyitk.
   700.  gulalegyitk te kcenecal = 600 + 100.
   800.  tqalpqtalegyitk.
   900.  tqalpqtalegyitk te kcenecal = 800 + 100.
  1000.  k'pal.

To the unobservant eye this scale would certainly appear to contain no more
than a trace of the vigesimal in its structure. But Dr. Boas, who is one of
the most careful and accurate of investigators, says in his comment on this
system: "It will be seen at once that this system is quinary-vigesimal....
In 20 we find the word _gyat_, man. The hundreds are identical with the
numerals used in counting men (see p. 87), and then the quinary-vigesimal
system is most evident."


  RIO NORTE INDIANS.[362]

    20.  taiguaco.
    30.  taiguaco co juyopamauj ajte = 20 + 2 × 5.
    40.  taiguaco ajte = 20 × 2.
    50.  taiguaco ajte co juyopamauj ajte = 20 × 2 + 5 × 2.


  CARIBS OF ESSIQUIBO, GUIANA

    10.  oween-abatoro.
    20.  owee-carena   = 1 person.
    40.  oko-carena    = 2 persons.
    60.  oroowa-carena = 3 persons.


  OTOMI

    10.  ra-tta.
    20.  na-te.
    30.  na-te-m'a-ratta = 20 + 10.
    40.  yo-te           = 2 × 30.
    50.  yote-m'a-ratta  = 2 × 20 + 10.
    60.  hiu-te          = 3 × 20.
    70.  hiute-m'a-ratta = 3 × 20 + 10.
    80.  gooho-rate      = 4 × 20.
    90.  gooho-rate-m'a ratta = 4 × 20 + 10.
   100.  cytta-te        = 5 × 20,
          or nanthebe    = 1 × 100.


  MAYA, YUCATAN.[363]

           1.  hun.
          10.  lahun  = it is finished.
          20.  hunkal = a measure, or more correctly, a fastening together.
          30.  lahucakal   = 40 - 10?
          40.  cakal       = 2 × 20.
          50.  lahuyoxkal  = 60 - 10.
          60.  oxkal       = 3 × 20.
          70.  lahucankal  = 80 - 10.
          80.  cankal      = 4 × 20.
          90.  lahuyokal   = 100 - 10.
         100.  hokal       = 5 × 20.
         110.  lahu uackal = 120 - 10.
         120.  uackal      = 6 × 20.
         130.  lahu uuckal = 140 - 10.
         140.  uuckal      = 7 × 20.
         200.  lahuncal    = 10 × 20.
         300.  holhukal    = 15 × 20.
         400.  hunbak      = 1 tying around.
         500.  hotubak.
         600.  lahutubak
         800.  calbak = 2 × 400.
         900.  hotu yoxbak.
        1000.  lahuyoxbak.
        1200.  oxbak = 3 × 400.
        2000.  capic (modern).
        8000.  hunpic = 1 sack.
      16,000.  ca pic (ancient).
     160,000.  calab = a filling full
   3,200,000.  kinchil.
  64,000,000.  hunalau.

In the Maya scale we have one of the best and most extended examples of
vigesimal numeration ever developed by any race. To show in a more striking
and forcible manner the perfect regularity of the system, the following
tabulation is made of the various Maya units, which will correspond to the
"10 units make one ten, 10 tens make one hundred, 10 hundreds make one
thousand," etc., which old-fashioned arithmetic compelled us to learn in
childhood. The scale is just as regular by twenties in Maya as by tens in
English. It is[364]

  20 hun     = 1 kal          =         20.
  20 kal     = 1 bak          =        400.
  20 bak     = 1 pic          =       8000.
  20 pic     = 1 calab        =    160,000.
  20 calab   = 1 { kinchil  } =  3,200,000.
                 { tzotzceh }
  20 kinchil = 1 alau         = 64,000,000.

The original meaning of _pic_, given in the scale as "a sack," was rather
"a short petticoat, somtimes used as a sack." The word _tzotzceh_ signified
"deerskin." No reason can be given for the choice of this word as a
numeral, though the appropriateness of the others is sufficiently manifest.
No evidence of digital numeration appears in the first 10 units, but,
judging from the almost universal practice of the Indian tribes of both
North and South America, such may readily have been the origin of Maya
counting. Whatever its origin, it certainly expanded and grew into a system
whose perfection challenges our admiration. It was worthy of the splendid
civilization of this unfortunate race, and, through its simplicity and
regularity, bears ample testimony to the intellectual capacity which
originated it.

The only example of vigesimal reckoning which is comparable with that of
the Mayas is the system employed by their northern neighbours, the Nahuatl,
or, as they are more commonly designated, the Aztecs of Mexico. This system
is quite as pure and quite as simple as the Maya, but differs from it in
some important particulars. In its first 20 numerals it is quinary (see p.
141), and as a system must be regarded as quinary-vigesimal. The Maya scale
is decimal through its first 20 numerals, and, if it is to be regarded as a
mixed scale, must be characterized as decimal-vigesimal. But in both these
instances the vigesimal element preponderates so strongly that these, in
common with their kindred number systems of Mexico, Yucatan, and Central
America, are always thought of and alluded to as vigesimal scales. On
account of its importance, the Nahuatl system[365] is given in fuller
detail than most of the other systems I have made use of.

          10.  matlactli = 2 hands.
          20.  cempoalli = 1 counting.
          21.  cempoalli once  = 20-1.
          22.  cempoalli omome = 20-2.
          30.  cempoalli ommatlactli      = 20-10.
          31.  cempoalli ommatlactli once = 20-10-1.
          40.  ompoalli               = 2 × 20.
          50.  ompoalli ommatlactli   = 40-10.
          60.  eipoalli, or epoalli,  = 3 × 20.
          70.  epoalli ommatlactli    = 60-10.
          80.  nauhpoalli             = 4 × 20.
          90.  nauhpoalli ommatlactli = 90-10.
         100.  macuilpoalli    = 5 × 20.
         120.  chiquacempoalli = 6 × 20.
         140.  chicompoalli    = 7 × 20.
         160.  chicuepoalli    = 8 × 20.
         180.  chiconauhpoalli = 9 × 20.
         200.  matlacpoalli    = 10 × 20.
         220.  matlactli oncempoalli  = 11 × 20.
         240.  matlactli omompoalli   = 12 × 20.
         260.  matlactli omeipoalli   = 13 × 20.
         280.  matlactli onnauhpoalli = 14 × 20.
         300.  caxtolpoalli           = 15 × 20.
         320.  caxtolli oncempoalli.
         399.  caxtolli onnauhpoalli ipan caxtolli onnaui = 19 × 20 + 19.
         400.  centzontli = 1 bunch of grass, or 1 tuft of hair.
         800.  ometzontli = 2 × 400.
        1200.  eitzontli  = 3 × 400.
        7600.  caxtolli onnauhtzontli = 19 × 400.
        8000.  cenxiquipilli, or cexiquipilli.
     160,000.  cempoalxiquipilli      = 20 × 8000.
   3,200,000.  centzonxiquipilli      = 400 × 8000.
  64,000,000.  cempoaltzonxiquipilli  = 20 × 400 × 8000.

Up to 160,000 the Nahuatl system is as simple and regular in its
construction as the English. But at this point it fails in the formation of
a new unit, or rather in the expression of its new unit by a simple word;
and in the expression of all higher numbers it is forced to resort in some
measure to compound terms, just as the English might have done had it not
been able to borrow from the Italian. The higher numeral terms, under such
conditions, rapidly become complex and cumbersome, as the following
analysis of the number 1,279,999,999 shows.[366] The analysis will be
readily understood when it is remembered that _ipan_ signifies plus.
_Caxtolli onnauhpoaltzonxiquipilli ipan caxtolli onnauhtzonxiquipilli ipan
caxtolli onnauhpoalxiquipilli ipan caxtolli onnauhxiquipilli ipan caxtolli
onnauhtzontli ipan caxtolli onnauhpoalli ipan caxtolli onnaui;_ _i.e._
1,216,000,000 + 60,800,000  + 3,040,000 + 152,000 + 7600 + 380 + 19. To
show the compounding which takes place in the higher numerals, the analysis
may be made more literally, thus:  + (15 + 4) × 400 × 800 + (15 + 4) × 20 ×
8000 + (15  + 4) × 8000 + (15 + 4) × 400 + (15 + 4) × 20 + 15  + 4. Of
course this resolution suffers from the fact that it is given in digits
arranged in accordance with decimal notation, while the Nahuatl numerals
express values by a base twice as great. This gives the effect of a
complexity and awkwardness greater than really existed in the actual use of
the scale. Except for the presence of the quinary element the number just
given is really expressed with just as great simplicity as it could be in
English words if our words "million" and "billion" were replaced by
"thousand thousand" and "thousand thousand thousand." If Mexico had
remained undisturbed by Europeans, and science and commerce had been left
to their natural growth and development, uncompounded words would
undoubtedly have been found for the higher units, 160,000, 3,200,000, etc.,
and the system thus rendered as simple as it is possible for a
quinary-vigesimal system to be.

Other number scales of this region are given as follows:


  HUASTECA.[367]

    10.  laluh.
    20.  hum-inic       = 1 man.
    30.  hum-inic-lahu  = 1 man 10.
    40.  tzab-inic      = 2 men.
    50.  tzab-inic-lahu = 2 men 10.
    60.  ox-inic        = 3 men.
    70.  ox-inic-lahu   = 3 men 10.
    80.  tze-tnic       = 4 men.
    90.  tze-ynic-kal-laluh = 4 men and 10.
   100.  bo-inic        = 5 men.
   200.  tzab-bo-inic   = 2 × 5 men.
   300.  ox-bo-inic     = 3 × 5 men.
   400.  tsa-bo-inic    = 4 × 5 men.
   600.  acac-bo-inic   = 6 × 5 men.
   800.  huaxic-bo-inic = 8 × 5 men.
  1000.  xi.
  8000.  huaxic-xi = 8-1000.

The essentially vigesimal character of this system changes in the formation
of some of the higher numerals, and a suspicion of the decimal enters. One
hundred is _boinic_, 5 men; but 200, instead of being simply _lahuh-inic_,
10 men, is _tsa-bo-inic_, 2 × 100, or more strictly, 2 times 5 men.
Similarly, 300 is 3 × 100, 400 is 4 × 100, etc. The word for 1000 is simple
instead of compound, and the thousands appear to be formed wholly on the
decimal base. A comparison of this scale with that of the Nahuatl shows how
much inferior it is to the latter, both in simplicity and consistency.


  TOTONACO.[368]

    10.  cauh.
    20.  puxam.
    30.  puxamacauh    = 20 + 10.
    40.  tipuxam       = 2 × 20.
    50.  tipuxamacauh  = 40 + 10.
    60.  totonpuxam    = 3 × 20.
   100.  quitziz puxum = 5 × 20.
   200.  copuxam       = 10 × 20.
   400.  tontaman.
  1000.  titamanacopuxam = 2 × 400 + 200.

The essential character of the vigesimal element is shown by the last two
numerals. _Tontamen_, the square of 20, is a simple word, and 1000 is, as
it should be, 2 times 400, plus 200. It is most unfortunate that the
numeral for 8000, the cube of 20, is not given.


  CORA.[369]

    10.  tamoamata.
    20.  cei-tevi.
    30.  ceitevi apoan tamoamata = 20 + 10.
    40.  huapoa-tevi  = 2 × 20.
    60.  huaeica-tevi = 3 × 20.
   100.  anxu-tevi    = 5 × 20.
   400.  ceitevi-tevi = 20 × 20.

Closely allied with the Maya numerals and method of counting are those of
the Quiches of Guatemala. The resemblance is so obvious that no detail in
the Quiche scale calls for special mention.


  QUICHE.[370]

    10.  lahuh.
    20.  hu-uinac       = 1 man.
    30.  hu-uinac-lahuh = 20 + 10.
    40.  ca-uinac       = 2 men.
    50.  lahu-r-ox-kal  = -10 + 3 × 20.
    60.  ox-kal         = 3 × 20.
    70.  lahu-u-humuch  = -10 + 80.
    80.  humuch.
    90.  lahu-r-ho-kal  = -10 + 100.
   100.  hokal.
  1000.  o-tuc-rox-o-kal.

Among South American vigesimal systems, the best known is that of the
Chibchas or Muyscas of the Bogota region, which was obtained at an early
date by the missionaries who laboured among them. This system is much less
extensive than that of some of the more northern races; but it is as
extensive as almost any other South American system with the exception of
the Peruvian, which was, however, a pure decimal system. As has already
been stated, the native races of South America were, as a rule, exceedingly
deficient in regard to the number sense. Their scales are rude, and show
great poverty, both in formation of numeral words and in the actual extent
to which counting was carried. If extended as far as 20, these scales are
likely to become vigesimal, but many stop far short of that limit, and no
inconsiderable number of them fail to reach even 5. In this respect we are
reminded of the Australian scales, which were so rudimentary as really to
preclude any proper use of the word "system" in connection with them.
Counting among the South American tribes was often equally limited, and
even less regular. Following are the significant numerals of the scale in
question:


  CHIBCHA, OR MUYSCA.[371]

    10.  hubchibica.
    20.  quihica ubchihica = thus says the foot, 10 = 10-10,
          or gueta         = house.
    30.  guetas asaqui ubchihica = 20 + 10.
    40.  gue-bosa    = 20 × 2.
    60.  gue-mica    = 20 × 3.
    80.  gue-muyhica = 20 × 4.
   100.  gue-hisca   = 20 × 5.


  NAGRANDA.[372]

    10.  guha.
    20.  dino.
    30.  'badiñoguhanu = 20 + 10.
    40.  apudiño       = 2 × 20.
    50.  apudiñoguhanu = 2 × 20 + 10.
    60.  asudiño       = 3 × 20.
    70.  asudiñoguhanu = 3 × 20 + 10.
    80.  acudiño       = 4 × 20.
    90.  acudiñoguhanu = 4 × 20 + 10.
   100.  huisudiño     = 5 × 20,
          or guhamba   = great 10.
   200.  guahadiño     = 10 × 20.
   400.  diñoamba      = great 20.
  1000.  guhaisudiño   = 10 × 5 × 20.
  2000.  hisudiñoamba  = 5 great 20's.
  4000.  guhadiñoamba  = 10 great 20's.

In considering the influence on the manners and customs of any people which
could properly be ascribed to the use among them of any other base than 10,
it must not be forgotten that no races, save those using that base, have
ever attained any great degree of civilization, with the exception of the
ancient Aztecs and their immediate neighbours, north and south. For reasons
already pointed out, no highly civilized race has ever used an exclusively
quinary system; and all that can be said of the influence of this mode of
counting is that it gives rise to the habit of collecting objects in groups
of five, rather than of ten, when any attempt is being made to ascertain
their sum. In the case of the subsidiary base 12, for which the Teutonic
races have always shown such a fondness, the dozen and gross of commerce,
the divisions of English money, and of our common weights and measures are
probably an outgrowth of this preference; and the Babylonian base, 60, has
fastened upon the world forever a sexagesimal method of dividing time, and
of measuring the circumference of the circle.

The advanced civilization attained by the races of Mexico and Central
America render it possible to see some of the effects of vigesimal
counting, just as a single thought will show how our entire lives are
influenced by our habit of counting by tens. Among the Aztecs the universal
unit was 20. A load of cloaks, of dresses, or other articles of convenient
size, was 20. Time was divided into periods of 20 days each. The armies
were numbered by divisions of 8000;[373] and in countless other ways the
vigesimal element of numbers entered into their lives, just as the decimal
enters into ours; and it is to be supposed that they found it as useful and
as convenient for all measuring purposes as we find our own system; as the
tradesman of to-day finds the duodecimal system of commerce; or as the
Babylonians of old found that singularly curious system, the sexagesimal.
Habituation, the laws which the habits and customs of every-day life impose
upon us, are so powerful, that our instinctive readiness to make use of any
concept depends, not on the intrinsic perfection or imperfection which
pertains to it, but on the familiarity with which previous use has invested
it. Hence, while one race may use a decimal, another a quinary-vigesimal,
and another a sexagesimal scale, and while one system may actually be
inherently superior to another, no user of one method of reckoning need
ever think of any other method as possessing practical inconveniences, of
which those employing it are ever conscious. And, to cite a single instance
which illustrates the unconscious daily use of two modes of reckoning in
one scale, we have only to think of the singular vigesimal fragment which
remains to this day imbedded in the numeral scale of the French. In
counting from 70 to 100, or in using any number which lies between those
limits, no Frenchman is conscious of employing a method of numeration less
simple or less convenient in any particular, than when he is at work with
the strictly decimal portions of his scale. He passes from the one style of
counting to the other, and from the second back to the first again,
entirely unconscious of any break or change; entirely unconscious, in fact,
that he is using any particular system, except that which the daily habit
of years has made a part himself.

Deep regret must be felt by every student of philology, that the primitive
meanings of simple numerals have been so generally lost. But, just as the
pebble on the beach has been worn and rounded by the beating of the waves
and by other pebbles, until no trace of its original form is left, and
until we can say of it now only that it is quartz, or that it is diorite,
so too the numerals of many languages have suffered from the attrition of
the ages, until all semblance of their origin has been lost, and we can say
of them only that they are numerals. Beyond a certain point we can carry
the study neither of number nor of number words. At that point both the
mathematician and the philologist must pause, and leave everything beyond
to the speculations of those who delight in nothing else so much as in pure
theory.




THE END.





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.
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.





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.
Bonzé, 151.
Bororo, 23.
Botocudo, 22, 31, 48, 71.
Bourke, 108.
Boyne River, 24.
Brazil, 2, 7, 30, 174, 195.
Bretagne (Breton), 120, 129, 181, 182.
British Columbia, 45, 46, 65, 86, 88, 89, 112, 113, 195.
Bullom, 147.
Bunch, 64.
Burnett River, 112.
Bushman, 28, 31.
Butong, 93.

Caddoe, 162.
Cahuillo, 165.
Calculating machine, 19.
Campa, 22.
Canada, 29, 53, 54, 86, 195.
Canaque, 142, 144.
Caraja, 23.
Carib, 166, 167, 199.
Carnarvon, 35, 36.
Carrier, 86.
Carthaginian, 179.
Caucasus, 188.
Cayriri (see Kiriri), 79.
Cayubaba (Cayubabi), 84, 167.
Celtic, 40, 169, 179, 181, 190.
Cely, Mom, 9.
Central America, 29, 69, 79, 121, 131, 195, 201, 208.
Ceylon, 28.
Chaco, 22.
Champion Bay, 109.
Charles XII., 132.
Cheyenne, 62.
Chibcha, 206.
China (Chinese), 40, 131.
Chippeway, 62, 159, 162.
Chiquito, 2, 6, 21, 40, 71, 76.
Choctaw, 65, 85, 162.
Chunsag, 189.
Circassia, 190.
Cobeu, 174.
Cochin China, 15.
Columbian, 113.
Comanche, 29, 83.
Conibo, 23.
Cooper's Creek, 108.
Cora, 166.
Cotoxo, 111.
Cowrie, 64, 70, 71.
Cree, 91.
Crocker Island, 107.
Crow, 3, 4, 92.
Crusoe, Robinson, 7.
Curetu, 111.

Dahomey, 71.
Dakota, 81, 91, 92.
Danish, 30, 46, 129, 183.
Darnley Islands, 24.
Delaware, 91, 160.
Demara, 4, 6.
Déné, 86.
Dido, 189.
Dinka, 136, 147.
Dippil, 107.
Division, 19.
Dravidian, 104, 193.
Dual number, 75.
Duluth, 34.
Duodecimal, chap. v.
Dutch, 129.

Eaw, 24.
Ebon, 152.
Efik, 148, 185.
Encabellada, 22.
Encounter Bay, 108.
Ende, 68, 152.
English, 28, 38-44, 60, 81, 85, 89, 118, 123, 124, 129, 183, 200, 203, 208.
Eromanga, 96, 136, 154.
Eskimo, 16, 30, 31, 32, 36, 48, 51, 52, 54, 61, 64, 83, 137, 157, 159, 195,
   196.
Essequibo, 166.
Europe (European), 27, 39, 168, 169, 179, 182, 183, 185, 204.
Eye, 14, 97.
Eyer's Sand Patch, 26.
Ewe, 64, 91.

Fall, 163.
Fate, 138, 155.
Fatuhiva, 130.
Feloop, 145.
Fernando Po, 150.
Fiji, 96.
Finger pantomime, 10, 23, 29, 67, 82.
Fingoe, 33.
Fist, 16, 59, 72.
Flinder's River, 24.
Flores, 68, 152.
Forefinger, 12, 15, 16, 54, 61, 91, 113.
Foulah, 147.
Fourth finger, 18.
Frazer's Island, 108.
French, 40, 41, 124, 129, 181, 182, 209.
Fuegan, 22.

Gaelic, 180.
Galibi, 138.
Gaul, 169, 182.
Georgia, 189.
German, 38-43, 129, 183.
Gesture, 18, 59.
Gola, 151.
Golo, 146.
Gonn Station, 110.
Goth, 169.
Greek, 86, 129, 168, 169.
Green Island, 45.
Greenland, 29, 52, 80, 158.
Guachi, 23, 31.
Guarani, 55, 138.
Guatemala, 205.
Guato, 142.
Guaycuru, 22.
Gudang, 24.

Haida, 112.
Hawaii, 113, 114, 116, 117.
Head, 71.
Heap, 8, 9, 25, 70, 77, 100.
Hebrew, 86, 89, 95.
Heiltsuk, 65, 88, 163.
Herero, 150.
Hervey Islands, 118.
Hidatsa, 80, 91.
Hill End, 109.
Himalaya, 193.
Hottentot, 80, 92.
Huasteca, 204.
Hudson's Bay, 48, 61.
Hun, 169.
Hunt, Leigh, 33.

Ibo, 185.
Icelandic, 129, 183.
Illinois, 91.
Index finger, 11, 14.
India, 96, 112, 131.
Indian, 8, 10, 13, 16, 17, 19, 32, 36, 54, 55, 59, 62, 65, 66, 79, 80, 82,
   83, 89, 90, 98, 105, 112, 171, 201.
Indian Ocean, 63, 193.
Indo-European, 76.
Irish, 129, 180.
Italian, 39, 80, 124, 129, 203.

Jajowerong, 156.
Jallonkas, 146.
Jaloff, 146.
Japanese, 40, 86, 89, 93-95.
Java, 93, 120.
Jiviro, 61, 136.
Joints of fingers, 7, 18, 173.
Juri, 79.

Kamassin, 130.
Kamilaroi, 27, 107, 112.
Kamtschatka, 75, 157.
Kanuri, 136, 149.
Karankawa, 68.
Karen, 112.
Keppel Bay, 24.
Ki-Nyassa, 150.
Kiriri, 55, 138, 139, 167.
Kissi, 145.
Ki-Swahili, 42.
Ki-Yau, 150.
Klamath, 58, 59.
Knot, 7, 9, 19, 40, 93, 115.
Kolyma, 75.
Kootenay, 65.
Koriak, 75.
Kredy, 149.
Kru, 146.
Ku-Mbutti, 78.
Kunama, 151.
Kuri, 188.
Kusaie, 78, 80.
Kwakiutl, 45.

Labillardière, 85.
Labrador, 29.
Lake Kopperamana, 107.
Latin, 40, 44, 76, 81, 86, 124, 128, 168, 169, 181, 182.
Lazi, 189.
Left hand, 10-17, 54.
Leper's Island, 16.
Leptscha, 193.
Lifu, 143.
Little finger, 10-18, 48, 54, 61, 91.
Logone, 186.
London, 124.
Lower California, 29.
Luli, 118.
Lutuami, 164.

Maba, 80.
Macassar, 93.
Machine, Calculating, 19, 20.
Mackenzie River, 157.
Macuni, 174.
Madagascar, 8, 9.
Maipures, 15, 56.
Mairassis, 10.
Malagasy, 83, 95.
Malanta, 96.
Malay, 8, 45, 90, 93, 170.
Mallicolo, 152.
Manadu, 93.
Mandingo, 186.
Mangareva, 114.
Manx, 180.
Many, 2, 21-23, 25, 28, 100.
Maori, 64, 93, 122.
Marachowie, 26.
Maré, 84.
Maroura, 106.
Marquesas, 93, 114, 115.
Marshall Islands, 122, 152.
Massachusetts, 91, 159.
Mathematician, 2, 3, 35, 102, 127, 210.
Matibani, 151.
Matlaltzinca, 166.
Maya, 45, 46, 199, 205.
Mbayi, 111.
Mbocobi, 22.
Mbousha, 66.
Melanesia, 16, 22, 28, 84, 95.
Mende, 186.
Mexico, 29, 195, 201, 204, 208.
Miami, 91.
Micmac, 90, 160.
Middle finger, 12, 15, 62.
Mille, 122.
Minnal Yungar, 26.
Minsi, 162.
Mississaga, 44, 92.
Mississippi, 125.
Mocobi, 119.
Mohegan, 91.
Mohican, 172.
Mokko, 149.
Molele, 164.
Moneroo, 109.
Mongolian, 8.
Montagnais, 53, 54, 175.
Moree, 24.
Moreton Bay, 108.
Mort Noular, 107.
Mosquito, 69, 70, 121.
Mota, 95, 153.
Mpovi, 152.
Multiplication, 19, 33, 40, 43, 59.
Mundari, 193.
Mundo, 186.
Muralug, 17.
Murray River, 106, 109.
Muysca, 206.

Nagranda, 207.
Nahuatl, 141, 144, 177, 201, 205.
Nakuhiva, 116, 130.
Negro, 8, 9, 15, 29, 184.
Nengone, 63, 136.
New, 128-130.
New Caledonia, 154.
New Granada, 195.
New Guinea, 10, 152.
New Hebrides, 155.
New Ireland, 45.
New Zealand, 123.
Nez Perces, 65, 158.
Ngarrimowro, 110.
Niam Niam, 64, 136.
Nicaragua, 80.
Nicobar, 78, 193.
Nightingale, 4.
Nootka, 163, 198.
Norman River, 24.
North America, 28, 82, 171, 173, 176, 194, 201.
Notch, 7, 9, 93.
Numeral frame, 19.
Nupe, 149, 186.
Nusqually, 96.

Oceania, 115, 176.
Octonary, chap. v.
Odessa, 34.
Ojibwa, 84, 159.
Okanaken, 88.
Omaha, 161.
Omeo, 110.
Oregon, 58, 195.
Orejone, 23.
Orinoco, 54, 56, 195.
Ostrich, 71, 72.
Otomac, 15.
Otomi, 165, 199.
Ottawa, 159.
Oyster Bay, 79.

Pacific, 29, 113, 116, 117, 131.
Palm (of the hand), 12, 14, 15.
Palm Island, 156.
Pama, 136, 155.
Pampanaga, 66.
Papaa, 148.
Paraguay, 55, 71, 118, 195.
Parana, 119.
Paris, 182.
Pawnee, 91, 121, 162.
Pebble, 7-9, 19, 40, 93, 179.
Peno, 2.
Peru (Peruvian), 2, 22, 61, 206.
Philippine, 66.
Philology (Philologist), 128, 209, 210.
Phoenician, 90, 179.
Pigmy, 69, 70, 78.
Pikumbul, 57, 138.
Pines, Isle of, 153.
Pinjarra, 26.
Plenty, 25, 77.
Point Barrow, 30, 51, 64, 83, 137, 159.
Polynesia, 22, 28, 118, 130, 170.
Pondo, 33.
Popham Bay, 107.
Port Darwin, 109.
Port Essington, 24, 107.
Port Mackay, 26.
Port Macquarie, 109.
Puget Sound, 96.
Puri, 22, 92.

Quappa, 171, 172.
Quaternary, chap. v.
Queanbeyan, 24.
Quiche, 205.
Quichua, 61.

Rapid, 163.
Rarotonga, 114.
Richmond River, 109.
Right hand, 10-18, 54.
Right-handedness, 13, 14.
Ring finger, 15.
Rio Grande, 195.
Rio Napo, 22.
Rio Norte, 136, 199.
Russia (Russian), 30, 35.

Sahaptin, 158.
San Antonio, 136.
San Blas, 79, 80.
Sanskrit, 40, 92, 97, 128.
Sapibocone, 84, 167.
Saste (Shasta), 165.
Scratch, 7.
Scythian, 169.
Seed, 93.
Semitic, 89.
Senary, chap. v.
Sesake, 136, 155.
Several, 22.
Sexagesimal, 124, 208.
Shawnoe, 160.
Shell, 7, 19, 70, 93.
Shushwap, 88.
Siberia, 29, 30, 187, 190.
Sierra Leone, 83.
Sign language, 6.
Sioux, 83.
Slang, 124.
Slavonic, 40.
Snowy River, 110.
Soussou, 83, 147.
South Africa, 4, 15, 28.
South America, 2, 15, 22, 23, 27-29, 54, 57, 72, 76, 78, 79, 104, 110, 173,
   174, 194, 201, 206.
Spanish, 2, 23, 42.
Splint, 7.
Stick, 7, 179.
Stlatlumh, 88.
Streaky Bay, 26.
String, 7, 9, 64, 71.
Strong's Island, 78.
Subtraction, 19, 44-47.
Sunda, 120.
Sweden (Swedish), 129, 132, 183.

Tacona, 2.
Taensa, 44.
Tagala, 66.
Tahiti, 114.
Tahuata, 115.
Tama, 111.
Tamanac, 54, 135.
Tambi, 120.
Tanna, 154.
Tarascan, 165.
Tariana, 174.
Tasmania, 24, 27, 79, 104, 106.
Tawgy, 130.
Tchetchnia, 188.
Tchiglit, 157, 196.
Tembu, 33.
Temne, 148.
Ternary, chap. v.
Terraba, 172.
Teutonic, 40, 41, 43, 179, 181, 208.
Texas, 69.
Thibet, 96.
Thumb, 10-18, 54, 59, 61, 62, 113, 173.
Thusch, 189.
Ticuna, 168.
Timukua, 165.
Tlingit, 136, 163, 197.
Tobi, 156.
Tonga, 33, 85.
Torres, 17, 96, 104, 105.
Totonaco, 205.
Towka, 78.
Triton's Bay, 152.
Tschukshi, 156, 191.
Tsimshian, 86, 164, 198.
Tweed River, 26.

Uainuma, 122.
Udi, 188.
Uea, 67, 153.
United States, 29, 83, 195.
Upper Yarra, 110.
Ureparapara, 153.

Vaturana, 96.
Vedda, 28, 31, 76, 174.
Vei, 16, 147, 185.
Victoria, 156.
Vilelo, 60.

Waiclatpu, 164.
Wales (Welsh), 35, 46, 141, 144, 177, 180.
Wallachia, 121.
Warrego, 107, 109.
Warrior Island, 107.
Wasp, 5.
Watchandie, 29, 77.
Watji, 120.
Weedookarry, 24.
Wimmera, 107.
Winnebago, 85.
Wiraduroi, 27, 108.
Wirri-Wirri, 108.
Wokke, 112.
Worcester, Mass., Schools of, 11.

Yahua, 168.
Yaruro, 139.
Yengen, 154.
Yit-tha, 109.
Yoruba, 33, 47, 64, 70, 185.
Yucatan, 195, 201.
Yuckaburra, 26.

Zamuco, 55, 60, 138, 139.
Zapara, 111.
Zulu, 16, 62.
Zuñi, 13, 14, 48, 49, 53, 54, 60, 83, 137.





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. 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.






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