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.