Levi Leonard Conant - The Number Concept

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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 x 4. 8. sin-the'de = 2 x 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 Benue, 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. alacoetegladu = 1 hand.
10. catoegladu = 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 x 2.

BARI

5. kanat
10. puoek = 5 + 5?

KANURI

5. ugu.
10. megu = 2 x 5.

RIO NORTE AND SAN ANTONIO.[231]

5. juyopamauj.
10. juyopamauj ajte = 5 x 2.

API.[232]

5. lima.
10. lua-lima = 2 x 5.

ERROMANGO

5. suku-rim.
10. nduru-lim = 2 x 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 Zuni system, in which the first
2 fives are:

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

It will be noticed that the Zuni 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 oietonai = 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 x 5) x 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 x 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 x 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 x 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 x 5.
11. ca ko.
12. lue ko.
13. koeni ko.
14. eke ko.
15. koeni pi = 3 x 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 x 2.

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