Levi Leonard Conant - The Number Concept

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1. de = a going, _i.e._ a beginning. (Cf. the Zuni _toepinte_, 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, maen = 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 x 5)(2 x 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-subquaetche_, 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.

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