Levi Leonard Conant - The Number Concept

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

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

QUAPPA.

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

TERRABA.[325]

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

MOHICAN

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

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

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

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

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

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

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

CHAPTER VII.

THE VIGESIMAL SYSTEM.

In its ordinary development the quinary system is almost sure to merge into
either the decimal or the vigesimal system, and to form, with one or the
other or both of these, a mixed system of counting. In Africa, Oceanica,
and parts of North America, the union is almost always with the decimal
scale; while in other parts of the world the quinary and the vigesimal
systems have shown a decided affinity for each other. It is not to be
understood that any geographical law of distribution has ever been observed
which governs this, but merely that certain families of races have shown a
preference for the one or the other method of counting. These families,
disseminating their characteristics through their various branches, have
produced certain groups of races which exhibit a well-marked tendency, here
toward the decimal, and there toward the vigesimal form of numeration. As
far as can be ascertained, the choice of the one or the other scale is
determined by no external circumstances, but depends solely on the mental
characteristics of the tribes themselves. Environment does not exert any
appreciable influence either. Both decimal and vigesimal numeration are
found indifferently in warm and in cold countries; in fruitful and in
barren lands; in maritime and in inland regions; and among highly civilized
or deeply degraded peoples.

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

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

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

IRISH.[329]

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

GAELIC.[330]

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

WELSH.[331]

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

MANX.[332]

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

BRETON.[333]

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

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

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

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

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

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

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

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

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

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

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

AFFADEH.[338]

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

IBO.[339]

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

VEI.[340]

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

YORUBA.[341]

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

EFIK.[342]

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

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

BONGO.[343]

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

MENDE.[344]

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

NUPE.[345]

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

LOGONE.[346]

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

MUNDO.[347]

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

MANDINGO.[348]

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

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

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

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

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