Levi Leonard Conant - The Number Concept

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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. apaedyai = 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 x 2.
7. then tho ta = 3 x 2-1.
8. lwie tho = 4 x 2.
9. lwie tho ta = 4 x 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 x 40 + 10; 200 is _lima
tanaha_, 5 x 40; and 864,895 is 2 x 400,000 + 40,000 + 6 x 4000 + 2 x 400 +
2 x 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 x 4, 100
x 4, 1000 x 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:

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