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 × 2.
50. tikkumgassigokang = 20 × 2 + 10.
60. tikkumgakro = 20 × 3.
70. dungokrogokang = 20 × 3 + 10.
80. dukumgade = 20 × 4.
90. dukumgadegokang = 20 × 4 + 10.
100. miah (borrowed from the Arabs).
IBO.[339]
10. iri.
20. ogu.
30. ogu n-iri = 20 + 10,
or iri ato = 10 × 3.
40. ogu abuo = 20 × 2,
or iri anno = 10 × 4.
100. ogu ise = 20 × 5.
VEI.[340]
10. tan.
20. mo bande = a person finished.
30. mo bande ako tan = 20 + 10.
40. mo fera bande = 2 × 20.
100. mo soru bande = 5 persons finished.
YORUBA.[341]
10. duup.
20. ogu.
30. ogbo.
40. ogo-dzi = 20 × 2.
60. ogo-ta = 20 × 3.
80. ogo-ri = 20 × 4.
100. ogo-ru = 20 × 5.
120. ogo-fa = 20 × 6.
140. ogo-dze = 20 × 7.
160. ogo-dzo = 20 × 8, etc.
EFIK.[342]
10. duup.
20. edip.
30. edip-ye-duup = 20 + 10.
40. aba = 20 × 2.
60. ata = 20 × 3.
80. anan = 20 × 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 × 1.
40. mbaba gnorr = 20 × 2.
100. mbaba mui = 20 × 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 × 20.
50. arota.
60. sita = 3 × 20.
70. adoni.
80. sini = 4 × 20.
90. sini be-guwo = 80 + 10.
100. sisun = 5 × 20.
LOGONE.[346]
10. chkan.
20. tkam.
30. tkam ka chkan = 20 + 10.
40. tkam ksde = 20 × 2.
50. tkam ksde ka chkan = 40 + 10.
60. tkam gachkir = 20 × 3.
100. mia (from Arabic).
1000. debu.
MUNDO.[347]
10. nujorquoi.
20. tiki bere.
30. tiki bire nujorquoi = 20 + 10.
40. tiki borsa = 20 × 2.
50. tike borsa nujorquoi = 40 + 10.
MANDINGO.[348]
10. tang.
20. mulu.
30. mulu nintang = 20 + 10.
40. mulu foola = 20 × 2.
50. mulu foola nintang = 40 + 10.
60. mulu sabba = 20 × 3.
70. mulu sabba nintang = 60 + 10.
80. mulu nani = 20 × 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:
ABKHASIA.[349]
10. zpha-ba.
20. gphozpha = 2 × 10.
30. gphozphei zphaba = 20 + 10.
40. gphin-gphozpha = 2 × 20.
60. chin-gphozpha = 3 × 20.
80. phsin-gphozpha = 4 × 20.
100. sphki.
AVARI
10. antsh-go.
20. qo-go.
30. lebergo.
40. khi-qogo = 2 × 20.
50. khiqojalda antshgo = 40 + 10.
60. lab-qogo = 3 × 20.
70. labqojalda antshgo = 60 + 10.
80. un-qogo = 4 × 20.
100. nusgo.
KURI
10. tshud.
20. chad.
30. channi tshud = 20 + 10.
40. jachtshur.
50. jachtshurni tshud = 40 + 10.
60. put chad = 3 × 20.
70. putchanni tshud = 60 + 10.
80. kud-chad = 4 × 20.
90. kudchanni tshud = 80 + 10.
100. wis.
UDI
10. witsh.
20. qa.
30. sa-qo-witsh = 20 + 10.
40. pha-qo = 2 × 20.
50. pha-qo-witsh = 40 + 10.
60. chib-qo = 3 × 20.
70. chib-qo-witsh = 60 + 10.
80. bip-qo = 4 × 20.
90. bip-qo-witsh = 80 + 10.
100. bats.
1000. hazar (Persian).
TCHETCHNIA
10. ith.
20. tqa.
30. tqe ith = 20 + 10.
40. sauz-tqa = 2 × 20.
50. sauz-tqe ith = 40 + 10.
60. chuz-tqa = 3 × 20.
70. chuz-tqe ith = 60 + 10.
80. w-iez-tqa = 4 × 20.
90. w-iez-tqe ith = 80 + 10.
100. b'e.
1000. ezir (akin to Persian).
THUSCH
10. itt.
20. tqa.
30. tqa-itt = 20 + 10.
40. sauz-tq = 2 × 20.
50. sauz-tqa-itt = 40 + 10.
60. chouz-tq = 3 × 20.
70. chouz-tqa-itt = 60 + 10.
80. dhewuz-tq = 4 × 20.
90. dhewuz-tqa-itt = 80 + 10.
100. phchauz-tq = 5 × 20.
200. itsha-tq = 10 × 20.
300. phehiitsha-tq = 15 × 20.
1000. satsh tqauz-tqa itshatqa = 2 × 20 × 20 + 200.
GEORGIA
10. athi.
20. otsi.
30. ots da athi = 20 + 10.
40. or-m-otsi = 2 × 20.
50. ormots da athi = 40 + 10.
60. sam-otsi = 3 × 20.
70. samots da athi = 60 + 10.
80. othch-m-otsi = 4 × 20.
90. othmots da athi = 80 + 10.
100. asi.
1000. ath-asi = 10 × 100.
LAZI
10. wit.
20. öts.
30. öts do wit = 20 × 10.
40. dzur en öts = 2 × 20.
50. dzur en öts do wit = 40 + 10.
60. dzum en öts = 3 × 20.
70. dzum en öts do wit = 60 + 10.
80. otch-an-öts = 4 × 20.
100. os.
1000. silia (akin to Greek).
CHUNSAG.[350]
10. ants-go.
20. chogo.
30. chogela antsgo = 20 + 10.
40. kichogo = 2 × 20.
50. kichelda antsgo = 40 + 10.
60. taw chago = 3 × 20.
70. taw chogelda antsgo = 60 + 10.
80. uch' chogo = 4 × 20.
90. uch' chogelda antsgo.
100. nusgo.
1000. asargo (akin to Persian).
DIDO.[351]
10. zino.
20. ku.
30. kunozino.
40. kaeno ku = 2 × 20.
50. kaeno kuno zino = 40 + 10.
60. sonno ku = 3 × 20.
70. sonno kuno zino = 60 + 10.
80. uino ku = 4 × 20.
90. uino huno zino = 80 + 10.
100. bischon.
400. kaeno kuno zino = 40 × 10.
AKARI
10. entzelgu.
20. kobbeggu.
30. lowergu.
40. kokawu = 2 × 20.
50. kikaldanske = 40 + 10.
60. secikagu.
70. kawalkaldansku = 3 × 20 + 10.
80. onkuku = 4 × 20.
90. onkordansku = 4 × 20 + 10.
100. nosku.
1000. askergu (from Persian).
CIRCASSIA
10. psche.
20. to-tsch.
30. totsch-era-pschirre = 20 + 10.
40. ptl'i-sch = 4 × 10.
50. ptl'isch-era-pschirre = 40 + 10.
60. chi-tsch = 6 × 10.
70. chitsch-era-pschirre = 60 + 10.
80. toshitl = 20 × 4?
90. toshitl-era-pschirre = 80 + 10.
100. scheh.
1000. min (Tartar) or schi-psche = 100 × 10.
The last of these scales is an unusual combination of decimal and
vigesimal. In the even tens it is quite regularly decimal, unless 80 is of
the structure suggested above. On the other hand, the odd tens are formed
in the ordinary vigesimal manner. The reason for this anomaly is not
obvious. I know of no other number system that presents the same
peculiarity, and cannot give any hypothesis which will satisfactorily
account for its presence here. In nearly all the examples given the decimal
becomes the leading element in the formation of all units above 100, just
as was the case in the Celtic scales already noticed.
Among the northern tribes of Siberia the numeral scales appear to be ruder
and less simple than those just examined, and the counting to be more
consistently vigesimal than in any scale we have thus far met with. The two
following examples are exceedingly interesting, as being among the best
illustrations of counting by twenties that are to be found anywhere in the
Old World.
TSCHUKSCHI.[352]
10. migitken = both hands.
20. chlik-kin = a whole man.
30. chlikkin mingitkin parol = 20 + 10.
40. nirach chlikkin = 2 × 20.
100. milin chlikkin = 5 × 20.
200. mingit chlikkin = 10 × 20, _i.e._ 10 men.
1000. miligen chlin-chlikkin = 5 × 200, _i.e._ five (times) 10 men.
AINO.[353]
10. wambi.
20. choz.
30. wambi i-doehoz = 10 from 40.
40. tochoz = 2 × 20.
50. wambi i-richoz = 10 from 60.
60. rechoz = 3 × 20.
70. wambi [i?] inichoz = 10 from 80.
80. inichoz = 4 × 20.
90. wambi aschikinichoz = 10 from 100.
100. aschikinichoz = 5 × 20.
110. wambi juwanochoz = 10 from 120.
120. juwano choz = 6 × 20.
130. wambi aruwanochoz = 10 from 140.
140. aruwano choz = 7 × 20.
150. wambi tubischano choz = 10 from 160.
160. tubischano choz = 8 × 20.
170. wambi schnebischano choz = 10 from 180.
180. schnebischano choz = 9 × 20.
190. wambi schnewano choz = 10 from 200.
200. schnewano choz = 10 × 20.
300. aschikinichoz i gaschima chnewano choz = 5 × 20 + 10 × 20.
400. toschnewano choz = 2 × (10 × 20).
500. aschikinichoz i gaschima toschnewano choz = 100 + 400.
600. reschiniwano choz = 3 × 200.
700. aschikinichoz i gaschima reschiniwano choz = 100 + 600.
800. inischiniwano choz = 4 × 200.
900. aschikinichoz i gaschima inischiniwano choz = 100 + 800.
1000. aschikini schinewano choz = 5 × 200.
2000. wanu schinewano choz = 10 × (10 × 20).
This scale is in one sense wholly vigesimal, and in another way it is not
to be regarded as pure, but as mixed. Below 20 it is quinary, and, however
far it might be extended, this quinary element would remain, making the
scale quinary-vigesimal. But in another sense, also, the Aino system is not
pure. In any unmixed vigesimal scale the word for 400 must be a simple
word, and that number must be taken as the vigesimal unit corresponding to
100 in the decimal scale. But the Ainos have no simple numeral word for any
number above 20, forming all higher numbers by combinations through one or
more of the processes of addition, subtraction, and multiplication. The
only number above 20 which is used as a unit is 200, which is expressed
merely as 10 twenties. Any even number of hundreds, or any number of
thousands, is then indicated as being so many times 10 twenties; and the
odd hundreds are so many times 10 twenties, plus 5 twenties more. This
scale is an excellent example of the cumbersome methods used by uncivilized
races in extending their number systems beyond the ordinary needs of daily
life.
In Central Asia a single vigesimal scale comes to light in the following
fragment of the Leptscha scale, of the Himalaya region:[354]
10. kati.
40. kafali = 4 × 10,
or kha nat = 2 × 20.
50. kafano = 5 × 10,
or kha nat sa kati = 2 × 20 + 10.
100. gjo, or kat.
Further to the south, among the Dravidian races, the vigesimal element is
also found. The following will suffice to illustrate the number systems of
these dialects, which, as far as the material at hand shows, are different
from each other only in minor particulars:
MUNDARI.[355]
10. gelea.
20. mi hisi.
30. mi hisi gelea = 20 + 10.
40. bar hisi = 2 × 20.
60. api hisi = 3 × 20.
80. upun hisi = 4 × 20.
100. mone hisi = 5 × 20.
In the Nicobar Islands of the Indian Ocean a well-developed example of
vigesimal numeration is found. The inhabitants of these islands are so low
in the scale of civilization that a definite numeral system of any kind is
a source of some surprise. Their neighbours, the Andaman Islanders, it will
be remembered, have but two numerals at their command; their intelligence
does not seem in any way inferior to that of the Nicobar tribes, and one is
at a loss to account for the superior development of the number sense in
the case of the latter. The intercourse of the coast tribes with traders
might furnish an explanation of the difficulty were it not for the fact
that the numeration of the inland tribes is quite as well developed as that
of the coast tribes; and as the former never come in contact with traders
and never engage in barter of any kind except in the most limited way, the
conclusion seems inevitable that this is merely one of the phenomena of
mental development among savage races for which we have at present no
adequate explanation. The principal numerals of the inland and of the coast
tribes are:[356]
INLAND TRIBES COAST TRIBES
10. teya. 10. sham.
20. heng-inai. 20. heang-inai.
30. heng-inai-tain 30. heang-inai-tanai
= 20 + 5 (couples). = 20 + 5 (couples).
40. au-inai = 2 × 20. 40. an-inai = 2 × 20.
100. tain-inai = 5 × 20. 100. tanai-inai = 5 × 20.
200. teya-inai = 10 × 20. 200. sham-inai = 10 × 20.
300. teya-tain-inai 300. heang-tanai-inai
= (10 + 5) × 20. = (10 + 5) 20.
400. heng-teo. 400. heang-momchiama.
In no other part of the world is vigesimal counting found so perfectly
developed, and, among native races, so generally preferred, as in North and
South America. In the eastern portions of North America and in the extreme
western portions of South America the decimal or the quinary decimal scale
is in general use. But in the northern regions of North America, in western
Canada and northwestern United States, in Mexico and Central America, and
in the northern and western parts of South America, the unit of counting
among the great majority of the native races was 20. The ethnological
affinities of these races are not yet definitely ascertained; and it is no
part of the scope of this work to enter into any discussion of that
involved question. But either through contact or affinity, this form of
numeration spread in prehistoric times over half or more than half of the
western hemisphere. It was the method employed by the rude Eskimos of the
north and their equally rude kinsmen of Paraguay and eastern Brazil; by the
forest Indians of Oregon and British Columbia, and by their more southern
kinsmen, the wild tribes of the Rio Grande and of the Orinoco. And, most
striking and interesting of all, it was the method upon which were based
the numeral systems of the highly civilized races of Mexico, Yucatan, and
New Granada. Some of the systems obtained from the languages of these
peoples are perfect, extended examples of vigesimal counting, not to be
duplicated in any other quarter of the globe. The ordinary unit was, as
would be expected, "one man," and in numerous languages the words for 20
and man are identical. But in other cases the original meaning of that
numeral word has been lost; and in others still it has a signification
quite remote from that given above. These meanings will be noticed in
connection with the scales themselves, which are given, roughly speaking,
in their geographical order, beginning with the Eskimo of the far north.
The systems of some of the tribes are as follows:
ALASKAN ESKIMOS.[357]
10. koleet.
20. enuenok.
30. enuenok kolinik = 20 + 10.
40. malho kepe ak = 2 × 20.
50. malho-kepe ak-kolmik che pah ak to = 2 × 20 + 10.
60. pingi shu-kepe ak = 3 × 20.
100. tale ma-kepe ak = 5 × 20.
400. enue nok ke pe ak = 20 × 20.
TCHIGLIT.[358]
10. krolit.
20. kroleti, or innun = man.
30. innok krolinik-tchikpalik = man + 2 hands.
40. innum mallerok = 2 men.
50. adjigaynarmitoat = as many times 10 as the fingers of the hand.
60. innumipit = 3 men.
70. innunmalloeronik arveneloerit = 7 men?
80. innun pinatçunik arveneloerit = 8 men?
90. innun tcitamanik arveneloerit = 9 men?
100. itchangnerkr.
1000. itchangner-park = great 100.
The meanings for 70, 80, 90, are not given by Father Petitot, but are of
such a form that the significations seem to be what are given above. Only a
full acquaintance with the Tchiglit language would justify one in giving
definite meanings to these words, or in asserting that an error had been
made in the numerals. But it is so remarkable and anomalous to find the
decimal and vigesimal scales mingled in this manner that one involuntarily
suspects either incompleteness of form, or an actual mistake.
TLINGIT.[359]
10. djinkat = both hands?
20. tle ka = 1 man.
30. natsk djinkat = 3 × 10.
40. dak'on djinkat = 4 × 10.
50. kedjin djinkat = 5 × 10.
60. tle durcu djinkat = 6 × 10.
70. daqa durcu djinkat = 7 × 10.
80. natska durcu djinkat = 8 × 10.
90. gocuk durcu djinkat = 9 × 10.
100. kedjin ka = 5 men, or 5 × 20.
200. djinkat ka = 10 × 20.
300. natsk djinkat ka = 30 men.
400. dak'on djinkat ka = 40 men.
This scale contains a strange commingling of decimal and vigesimal
counting. The words for 20, 100, and 200 are clear evidence of vigesimal,
while 30 to 90, and the remaining hundreds, are equally unmistakable proof
of decimal, numeration. The word _ka_, man, seems to mean either 10 or 20;
a most unusual occurrence. The fact that a number system is partly decimal
and partly vigesimal is found to be of such frequent occurrence that this
point in the Tlingit scale need excite no special wonder. But it is
remarkable that the same word should enter into numeral composition under
such different meanings.
NOOTKA.[360]
10. haiu.
20. tsakeits.
30. tsakeits ic haiu = 20 + 10.
40. atlek = 2 × 20.
60. katstsek = 3 × 20.
80. moyek = 4 × 20.
100. sutc'ek = 5 × 20.
120. nop'ok = 6 × 20.
140. atlpok = 7 × 20.
160. atlakutlek = 8 × 20.
180. ts'owakutlek = 9 × 20.
200. haiuk = 10 × 20.
This scale is quinary-vigesimal, with no apparent decimal element in its
composition. But the derivation of some of the terms used is detected with
difficulty. In the following scale the vigesimal structure is still more
obscure.
TSIMSHIAN.[361]
10. gy'ap.
20. kyedeel = 1 man.
30. gulewulgy'ap.
40. t'epqadalgyitk, or tqalpqwulgyap.
50. kctoncwulgyap.
100. kcenecal.
200. k'pal.
300. k'pal te kcenecal = 200 + 100.
400. kyedal.
500. kyedal te kcenecal = 400 + 100.
600. gulalegyitk.
700. gulalegyitk te kcenecal = 600 + 100.
800. tqalpqtalegyitk.
900. tqalpqtalegyitk te kcenecal = 800 + 100.
1000. k'pal.
To the unobservant eye this scale would certainly appear to contain no more
than a trace of the vigesimal in its structure. But Dr. Boas, who is one of
the most careful and accurate of investigators, says in his comment on this
system: "It will be seen at once that this system is quinary-vigesimal....
In 20 we find the word _gyat_, man. The hundreds are identical with the
numerals used in counting men (see p. 87), and then the quinary-vigesimal
system is most evident."
RIO NORTE INDIANS.[362]
20. taiguaco.
30. taiguaco co juyopamauj ajte = 20 + 2 × 5.
40. taiguaco ajte = 20 × 2.
50. taiguaco ajte co juyopamauj ajte = 20 × 2 + 5 × 2.
CARIBS OF ESSIQUIBO, GUIANA
10. oween-abatoro.
20. owee-carena = 1 person.
40. oko-carena = 2 persons.
60. oroowa-carena = 3 persons.
OTOMI
10. ra-tta.
20. na-te.
30. na-te-m'a-ratta = 20 + 10.
40. yo-te = 2 × 30.
50. yote-m'a-ratta = 2 × 20 + 10.
60. hiu-te = 3 × 20.
70. hiute-m'a-ratta = 3 × 20 + 10.
80. gooho-rate = 4 × 20.
90. gooho-rate-m'a ratta = 4 × 20 + 10.
100. cytta-te = 5 × 20,
or nanthebe = 1 × 100.
MAYA, YUCATAN.[363]
1. hun.
10. lahun = it is finished.
20. hunkal = a measure, or more correctly, a fastening together.
30. lahucakal = 40 - 10?
40. cakal = 2 × 20.
50. lahuyoxkal = 60 - 10.
60. oxkal = 3 × 20.
70. lahucankal = 80 - 10.
80. cankal = 4 × 20.
90. lahuyokal = 100 - 10.
100. hokal = 5 × 20.
110. lahu uackal = 120 - 10.
120. uackal = 6 × 20.
130. lahu uuckal = 140 - 10.
140. uuckal = 7 × 20.
200. lahuncal = 10 × 20.
300. holhukal = 15 × 20.
400. hunbak = 1 tying around.
500. hotubak.
600. lahutubak
800. calbak = 2 × 400.
900. hotu yoxbak.
1000. lahuyoxbak.
1200. oxbak = 3 × 400.
2000. capic (modern).
8000. hunpic = 1 sack.
16,000. ca pic (ancient).
160,000. calab = a filling full
3,200,000. kinchil.
64,000,000. hunalau.
In the Maya scale we have one of the best and most extended examples of
vigesimal numeration ever developed by any race. To show in a more striking
and forcible manner the perfect regularity of the system, the following
tabulation is made of the various Maya units, which will correspond to the
"10 units make one ten, 10 tens make one hundred, 10 hundreds make one
thousand," etc., which old-fashioned arithmetic compelled us to learn in
childhood. The scale is just as regular by twenties in Maya as by tens in
English. It is[364]
20 hun = 1 kal = 20.
20 kal = 1 bak = 400.
20 bak = 1 pic = 8000.
20 pic = 1 calab = 160,000.
20 calab = 1 { kinchil } = 3,200,000.
{ tzotzceh }
20 kinchil = 1 alau = 64,000,000.
The original meaning of _pic_, given in the scale as "a sack," was rather
"a short petticoat, somtimes used as a sack." The word _tzotzceh_ signified
"deerskin." No reason can be given for the choice of this word as a
numeral, though the appropriateness of the others is sufficiently manifest.
No evidence of digital numeration appears in the first 10 units, but,
judging from the almost universal practice of the Indian tribes of both
North and South America, such may readily have been the origin of Maya
counting. Whatever its origin, it certainly expanded and grew into a system
whose perfection challenges our admiration. It was worthy of the splendid
civilization of this unfortunate race, and, through its simplicity and
regularity, bears ample testimony to the intellectual capacity which
originated it.
The only example of vigesimal reckoning which is comparable with that of
the Mayas is the system employed by their northern neighbours, the Nahuatl,
or, as they are more commonly designated, the Aztecs of Mexico. This system
is quite as pure and quite as simple as the Maya, but differs from it in
some important particulars. In its first 20 numerals it is quinary (see p.
141), and as a system must be regarded as quinary-vigesimal. The Maya scale
is decimal through its first 20 numerals, and, if it is to be regarded as a
mixed scale, must be characterized as decimal-vigesimal. But in both these
instances the vigesimal element preponderates so strongly that these, in
common with their kindred number systems of Mexico, Yucatan, and Central
America, are always thought of and alluded to as vigesimal scales. On
account of its importance, the Nahuatl system[365] is given in fuller
detail than most of the other systems I have made use of.
10. matlactli = 2 hands.
20. cempoalli = 1 counting.
21. cempoalli once = 20-1.
22. cempoalli omome = 20-2.
30. cempoalli ommatlactli = 20-10.
31. cempoalli ommatlactli once = 20-10-1.
40. ompoalli = 2 × 20.
50. ompoalli ommatlactli = 40-10.
60. eipoalli, or epoalli, = 3 × 20.
70. epoalli ommatlactli = 60-10.
80. nauhpoalli = 4 × 20.
90. nauhpoalli ommatlactli = 90-10.
100. macuilpoalli = 5 × 20.
120. chiquacempoalli = 6 × 20.
140. chicompoalli = 7 × 20.
160. chicuepoalli = 8 × 20.
180. chiconauhpoalli = 9 × 20.
200. matlacpoalli = 10 × 20.
220. matlactli oncempoalli = 11 × 20.
240. matlactli omompoalli = 12 × 20.
260. matlactli omeipoalli = 13 × 20.
280. matlactli onnauhpoalli = 14 × 20.
300. caxtolpoalli = 15 × 20.
320. caxtolli oncempoalli.
399. caxtolli onnauhpoalli ipan caxtolli onnaui = 19 × 20 + 19.
400. centzontli = 1 bunch of grass, or 1 tuft of hair.
800. ometzontli = 2 × 400.
1200. eitzontli = 3 × 400.
7600. caxtolli onnauhtzontli = 19 × 400.
8000. cenxiquipilli, or cexiquipilli.
160,000. cempoalxiquipilli = 20 × 8000.
3,200,000. centzonxiquipilli = 400 × 8000.
64,000,000. cempoaltzonxiquipilli = 20 × 400 × 8000.
Up to 160,000 the Nahuatl system is as simple and regular in its
construction as the English. But at this point it fails in the formation of
a new unit, or rather in the expression of its new unit by a simple word;
and in the expression of all higher numbers it is forced to resort in some
measure to compound terms, just as the English might have done had it not
been able to borrow from the Italian. The higher numeral terms, under such
conditions, rapidly become complex and cumbersome, as the following
analysis of the number 1,279,999,999 shows.[366] The analysis will be
readily understood when it is remembered that _ipan_ signifies plus.
_Caxtolli onnauhpoaltzonxiquipilli ipan caxtolli onnauhtzonxiquipilli ipan
caxtolli onnauhpoalxiquipilli ipan caxtolli onnauhxiquipilli ipan caxtolli
onnauhtzontli ipan caxtolli onnauhpoalli ipan caxtolli onnaui;_ _i.e._
1,216,000,000 + 60,800,000 + 3,040,000 + 152,000 + 7600 + 380 + 19. To
show the compounding which takes place in the higher numerals, the analysis
may be made more literally, thus: + (15 + 4) × 400 × 800 + (15 + 4) × 20 ×
8000 + (15 + 4) × 8000 + (15 + 4) × 400 + (15 + 4) × 20 + 15 + 4. Of
course this resolution suffers from the fact that it is given in digits
arranged in accordance with decimal notation, while the Nahuatl numerals
express values by a base twice as great. This gives the effect of a
complexity and awkwardness greater than really existed in the actual use of
the scale. Except for the presence of the quinary element the number just
given is really expressed with just as great simplicity as it could be in
English words if our words "million" and "billion" were replaced by
"thousand thousand" and "thousand thousand thousand." If Mexico had
remained undisturbed by Europeans, and science and commerce had been left
to their natural growth and development, uncompounded words would
undoubtedly have been found for the higher units, 160,000, 3,200,000, etc.,
and the system thus rendered as simple as it is possible for a
quinary-vigesimal system to be.
Other number scales of this region are given as follows:
HUASTECA.[367]
10. laluh.
20. hum-inic = 1 man.
30. hum-inic-lahu = 1 man 10.
40. tzab-inic = 2 men.
50. tzab-inic-lahu = 2 men 10.
60. ox-inic = 3 men.
70. ox-inic-lahu = 3 men 10.
80. tze-tnic = 4 men.
90. tze-ynic-kal-laluh = 4 men and 10.
100. bo-inic = 5 men.
200. tzab-bo-inic = 2 × 5 men.
300. ox-bo-inic = 3 × 5 men.
400. tsa-bo-inic = 4 × 5 men.
600. acac-bo-inic = 6 × 5 men.
800. huaxic-bo-inic = 8 × 5 men.
1000. xi.
8000. huaxic-xi = 8-1000.
The essentially vigesimal character of this system changes in the formation
of some of the higher numerals, and a suspicion of the decimal enters. One
hundred is _boinic_, 5 men; but 200, instead of being simply _lahuh-inic_,
10 men, is _tsa-bo-inic_, 2 × 100, or more strictly, 2 times 5 men.
Similarly, 300 is 3 × 100, 400 is 4 × 100, etc. The word for 1000 is simple
instead of compound, and the thousands appear to be formed wholly on the
decimal base. A comparison of this scale with that of the Nahuatl shows how
much inferior it is to the latter, both in simplicity and consistency.
TOTONACO.[368]
10. cauh.
20. puxam.
30. puxamacauh = 20 + 10.
40. tipuxam = 2 × 20.
50. tipuxamacauh = 40 + 10.
60. totonpuxam = 3 × 20.
100. quitziz puxum = 5 × 20.
200. copuxam = 10 × 20.
400. tontaman.
1000. titamanacopuxam = 2 × 400 + 200.
The essential character of the vigesimal element is shown by the last two
numerals. _Tontamen_, the square of 20, is a simple word, and 1000 is, as
it should be, 2 times 400, plus 200. It is most unfortunate that the
numeral for 8000, the cube of 20, is not given.
CORA.[369]
10. tamoamata.
20. cei-tevi.
30. ceitevi apoan tamoamata = 20 + 10.
40. huapoa-tevi = 2 × 20.
60. huaeica-tevi = 3 × 20.
100. anxu-tevi = 5 × 20.
400. ceitevi-tevi = 20 × 20.
Closely allied with the Maya numerals and method of counting are those of
the Quiches of Guatemala. The resemblance is so obvious that no detail in
the Quiche scale calls for special mention.
QUICHE.[370]
10. lahuh.
20. hu-uinac = 1 man.
30. hu-uinac-lahuh = 20 + 10.
40. ca-uinac = 2 men.
50. lahu-r-ox-kal = -10 + 3 × 20.
60. ox-kal = 3 × 20.
70. lahu-u-humuch = -10 + 80.
80. humuch.
90. lahu-r-ho-kal = -10 + 100.
100. hokal.
1000. o-tuc-rox-o-kal.
Among South American vigesimal systems, the best known is that of the
Chibchas or Muyscas of the Bogota region, which was obtained at an early
date by the missionaries who laboured among them. This system is much less
extensive than that of some of the more northern races; but it is as
extensive as almost any other South American system with the exception of
the Peruvian, which was, however, a pure decimal system. As has already
been stated, the native races of South America were, as a rule, exceedingly
deficient in regard to the number sense. Their scales are rude, and show
great poverty, both in formation of numeral words and in the actual extent
to which counting was carried. If extended as far as 20, these scales are
likely to become vigesimal, but many stop far short of that limit, and no
inconsiderable number of them fail to reach even 5. In this respect we are
reminded of the Australian scales, which were so rudimentary as really to
preclude any proper use of the word "system" in connection with them.
Counting among the South American tribes was often equally limited, and
even less regular. Following are the significant numerals of the scale in
question:
CHIBCHA, OR MUYSCA.[371]
10. hubchibica.
20. quihica ubchihica = thus says the foot, 10 = 10-10,
or gueta = house.
30. guetas asaqui ubchihica = 20 + 10.
40. gue-bosa = 20 × 2.
60. gue-mica = 20 × 3.
80. gue-muyhica = 20 × 4.
100. gue-hisca = 20 × 5.
NAGRANDA.[372]
10. guha.
20. dino.
30. 'badiñoguhanu = 20 + 10.
40. apudiño = 2 × 20.
50. apudiñoguhanu = 2 × 20 + 10.
60. asudiño = 3 × 20.
70. asudiñoguhanu = 3 × 20 + 10.
80. acudiño = 4 × 20.
90. acudiñoguhanu = 4 × 20 + 10.
100. huisudiño = 5 × 20,
or guhamba = great 10.
200. guahadiño = 10 × 20.
400. diñoamba = great 20.
1000. guhaisudiño = 10 × 5 × 20.
2000. hisudiñoamba = 5 great 20's.
4000. guhadiñoamba = 10 great 20's.
In considering the influence on the manners and customs of any people which
could properly be ascribed to the use among them of any other base than 10,
it must not be forgotten that no races, save those using that base, have
ever attained any great degree of civilization, with the exception of the
ancient Aztecs and their immediate neighbours, north and south. For reasons
already pointed out, no highly civilized race has ever used an exclusively
quinary system; and all that can be said of the influence of this mode of
counting is that it gives rise to the habit of collecting objects in groups
of five, rather than of ten, when any attempt is being made to ascertain
their sum. In the case of the subsidiary base 12, for which the Teutonic
races have always shown such a fondness, the dozen and gross of commerce,
the divisions of English money, and of our common weights and measures are
probably an outgrowth of this preference; and the Babylonian base, 60, has
fastened upon the world forever a sexagesimal method of dividing time, and
of measuring the circumference of the circle.
The advanced civilization attained by the races of Mexico and Central
America render it possible to see some of the effects of vigesimal
counting, just as a single thought will show how our entire lives are
influenced by our habit of counting by tens. Among the Aztecs the universal
unit was 20. A load of cloaks, of dresses, or other articles of convenient
size, was 20. Time was divided into periods of 20 days each. The armies
were numbered by divisions of 8000;[373] and in countless other ways the
vigesimal element of numbers entered into their lives, just as the decimal
enters into ours; and it is to be supposed that they found it as useful and
as convenient for all measuring purposes as we find our own system; as the
tradesman of to-day finds the duodecimal system of commerce; or as the
Babylonians of old found that singularly curious system, the sexagesimal.
Habituation, the laws which the habits and customs of every-day life impose
upon us, are so powerful, that our instinctive readiness to make use of any
concept depends, not on the intrinsic perfection or imperfection which
pertains to it, but on the familiarity with which previous use has invested
it. Hence, while one race may use a decimal, another a quinary-vigesimal,
and another a sexagesimal scale, and while one system may actually be
inherently superior to another, no user of one method of reckoning need
ever think of any other method as possessing practical inconveniences, of
which those employing it are ever conscious. And, to cite a single instance
which illustrates the unconscious daily use of two modes of reckoning in
one scale, we have only to think of the singular vigesimal fragment which
remains to this day imbedded in the numeral scale of the French. In
counting from 70 to 100, or in using any number which lies between those
limits, no Frenchman is conscious of employing a method of numeration less
simple or less convenient in any particular, than when he is at work with
the strictly decimal portions of his scale. He passes from the one style of
counting to the other, and from the second back to the first again,
entirely unconscious of any break or change; entirely unconscious, in fact,
that he is using any particular system, except that which the daily habit
of years has made a part himself.
Deep regret must be felt by every student of philology, that the primitive
meanings of simple numerals have been so generally lost. But, just as the
pebble on the beach has been worn and rounded by the beating of the waves
and by other pebbles, until no trace of its original form is left, and
until we can say of it now only that it is quartz, or that it is diorite,
so too the numerals of many languages have suffered from the attrition of
the ages, until all semblance of their origin has been lost, and we can say
of them only that they are numerals. Beyond a certain point we can carry
the study neither of number nor of number words. At that point both the
mathematician and the philologist must pause, and leave everything beyond
to the speculations of those who delight in nothing else so much as in pure
theory.
THE END.
INDEX OF AUTHORS.
Adam, L., 44, 159, 166, 175.
Armstrong, R.A., 180.
Aymonier, A., 156.
Bachofen, J.J., 131.
Balbi, A., 151.
Bancroft, H.H., 29, 47, 89, 93, 113, 199.
Barlow, H., 108.
Beauregard, O., 45, 83, 152.
Bellamy, E.W., 9.
Boas, F., 30, 45, 46, 65, 87, 88, 136, 163, 164, 171, 197, 198.
Bonwick, J., 24, 27, 107, 108.
Brinton, D.G., 2, 22, 46, 52, 57, 61, 111, 112, 140, 199, 200.
Burton, R.F., 37, 71.
Chamberlain, A.F., 45, 65, 93.
Chase, P.E., 99.
Clarke, H., 113.
Codrington, R.H., 16, 95, 96, 136, 138, 145, 153, 154.
Crawfurd, J., 89, 93, 130.
Curr, E.M., 24-27, 104, 107-110, 112.
Cushing, F.H., 13, 48.
De Flacourt, 8, 9.
De Quincey, T., 35.
Deschamps, M., 28.
Dobrizhoffer, M., 71.
Dorsey, J.O., 59.
Du Chaillu, P.B., 66, 67, 150, 151.
Du Graty, A.M., 138.
Ellis, A.A., 64, 91.
Ellis, R., 37, 142.
Ellis, W., 83, 119.
Erskine, J.E., 153, 154.
Flegel, R., 133.
Gallatin, A., 136, 159, 166, 171, 199, 204, 206, 208.
Galton, F., 4.
Gatschet, A.S., 58, 59, 68.
Gilij, F.S., 54.
Gill, W.W., 18, 118.
Goedel, M., 83, 147.
Grimm, J.L.C., 48.
Gröber, G., 182.
Guillome, J., 181.
Haddon, A.C., 18, 105.
Hale, H., 61, 65, 93, 114-116, 122, 130, 156, 163, 164, 171.
Hankel, H., 137.
Haumonté, J.D., 44.
Hervas, L., 170.
Humboldt, A. von, 32, 207.
Hyades, M., 22.
Kelly, J.W., 157, 196.
Kelly, J., 180.
Kleinschmidt, S., 52, 80.
Lang, J.D., 108.
Lappenberg, J.M., 127.
Latham, R.G., 24, 67, 107.
Leibnitz, G.W. von, 102, 103.
Lloyd, H.E., 7.
Long, C.C., 148, 186.
Long, S.H., 121.
Lubbock, Sir J., 3, 5.
Lull, E.P., 79.
Macdonald, J., 15.
Mackenzie, A., 26.
Man, E.H., 28, 194.
Mann, A., 47.
Marcoy, P. (Saint Cricq), 23, 168.
Mariner, A., 85.
Martius, C.F. von, 23, 79, 111, 122, 138, 142, 174.
Mason, 112.
Mill, J.S., 3.
Moncelon, M., 142.
Morice, A., 15, 86.
Müller, Fr., 10, 27, 28, 45, 48, 55, 56, 60, 63, 66, 69, 78, 80, 90, 108,
111, 121, 122, 130, 136, 139, 146-151, 156-158, 165-167, 185-187, 191,
193.
Murdoch, J., 30, 49,137.
Nystron, J.W., 132.
O'Donovan, J., 180.
Oldfield, A., 29, 77.
Olmos, A. de, 141.
Parisot, J., 44.
Park, M., 145-147.
Parry, W.E., 32.
Peacock, G., 8, 56, 84, 111, 118, 119, 154, 186.
Petitot, E., 53, 157, 196.
Pott, A.F., 50, 68, 92, 120, 145, 148, 149, 152, 157, 166, 182, 184, 189,
191, 205.
Pruner-Bey, 10, 104.
Pughe, W.O., 141.
Ralph, J., 125.
Ray, S.H., 45, 78, 80.
Ridley, W., 57.
Roth, H.L., 79.
Salt, H., 187.
Sayce, A.H., 75.
Schoolcraft, H.R., 66, 81, 83, 84, 159, 160.
Schröder, P., 90.
Schweinfurth, G., 143, 146, 149, 186, 187.
Simeon, R., 201.
Spix, J.B. von, 7.
Spurrell, W., 180.
Squier, G.E., 80, 207.
Stanley, H.M., 38, 42, 64, 69, 78, 150, 187.
Taplin, G., 106.
Thiel, B.A., 172.
Toy, C.H., 70.
Turner, G., 152, 154.
Tylor, E.B., 2, 3, 15, 18, 22, 63, 65, 78, 79, 81, 84, 97, 124.
Van Eys, J.W., 182.
Vignoli, T., 95.
Wallace, A.R., 174.
Wells, E.R., jr., 157, 196.
Whewell, W., 3.
Wickersham, J., 96.
Wiener, C., 22.
Williams, W.L., 123.
INDEX OF SUBJECTS.
Abacus, 19.
Abeokuta, 33.
Abipone, 71, 72.
Abkhasia, 188.
Aboker, 148.
Actuary, Life ins., 19.
Adaize, 162.
Addition, 19, 43, 46, 92.
Adelaide, 108.
Admiralty Islands, 45.
Affadeh, 184.
Africa (African), 9, 16, 28, 29, 32, 33, 38, 42, 47, 64, 66, 69, 78, 80,
91, 105, 120, 145, 170, 176, 184, 187.
Aino (Ainu), 45, 191.
Akra, 120.
Akari, 190.
Alaska, 157, 196.
Albania, 184.
Albert River, 26.
Aleut, 157.
Algonkin (Algonquin), 45, 92, 161.
Amazon, 23.
Ambrym, 136.
American, 10, 16, 19, 98, 105.
Andaman, 8, 15, 28, 31, 76, 174, 193.
Aneitum, 154.
Animal, 3, 6.
Anthropological, 21.
Apho, 133.
Api, 80, 136, 155.
Apinage, 111.
Arab, 170.
Arawak, 52-54, 135.
Arctic, 29.
Arikara, 46.
Arithmetic, 1, 5, 30, 33, 73, 93.
Aryan, 76, 128-130.
Ashantee, 145.
Asia (Asiatic), 28, 113, 131, 187.
Assiniboine, 66, 92.
Athapaskan,92.
Atlantic, 126.
Aurora, 155.
Australia (Australian), 2, 6, 19, 22, 24-30, 57, 58, 71, 75, 76, 84, 103,
105, 106, 110, 112, 118, 173, 206.
Avari, 188.
Aymara, 166.
Aztec, 63, 78, 83, 89, 93, 201, 207, 208.
Babusessé, 38.
Babylonian, 208.
Bagrimma, 148.
Bahnars, 15.
Bakairi, 111.
Balad, 67.
Balenque, 150.
Bambarese, 95.
Banks Islands, 16, 96, 153.
Barea, 151.
Bargaining, 18, 19, 32.
Bari, 136.
Barre, 174.
Basa, 146.
Basque, 40, 182.
Bellacoola, see Bilqula.
Belyando River, 109.
Bengal, Bay of, 28.
Benuë, 133.
Betoya, 57, 112, 135, 140.
Bilqula, 46, 164.
Binary, chap. v.
Binin, 149.
Bird-nesting, 5.
Bisaye, 90.
Bogota, 206.
Bolan, 120.
Bolivia, 2, 21.
Bongo, 143, 186.
Bonzé, 151.
Bororo, 23.
Botocudo, 22, 31, 48, 71.
Bourke, 108.
Boyne River, 24.
Brazil, 2, 7, 30, 174, 195.
Bretagne (Breton), 120, 129, 181, 182.
British Columbia, 45, 46, 65, 86, 88, 89, 112, 113, 195.
Bullom, 147.
Bunch, 64.
Burnett River, 112.
Bushman, 28, 31.
Butong, 93.
Caddoe, 162.
Cahuillo, 165.
Calculating machine, 19.
Campa, 22.
Canada, 29, 53, 54, 86, 195.
Canaque, 142, 144.
Caraja, 23.
Carib, 166, 167, 199.
Carnarvon, 35, 36.
Carrier, 86.
Carthaginian, 179.
Caucasus, 188.
Cayriri (see Kiriri), 79.
Cayubaba (Cayubabi), 84, 167.
Celtic, 40, 169, 179, 181, 190.
Cely, Mom, 9.
Central America, 29, 69, 79, 121, 131, 195, 201, 208.
Ceylon, 28.
Chaco, 22.
Champion Bay, 109.
Charles XII., 132.
Cheyenne, 62.
Chibcha, 206.
China (Chinese), 40, 131.
Chippeway, 62, 159, 162.
Chiquito, 2, 6, 21, 40, 71, 76.
Choctaw, 65, 85, 162.
Chunsag, 189.
Circassia, 190.
Cobeu, 174.
Cochin China, 15.
Columbian, 113.
Comanche, 29, 83.
Conibo, 23.
Cooper's Creek, 108.
Cora, 166.
Cotoxo, 111.
Cowrie, 64, 70, 71.
Cree, 91.
Crocker Island, 107.
Crow, 3, 4, 92.
Crusoe, Robinson, 7.
Curetu, 111.
Dahomey, 71.
Dakota, 81, 91, 92.
Danish, 30, 46, 129, 183.
Darnley Islands, 24.
Delaware, 91, 160.
Demara, 4, 6.
Déné, 86.
Dido, 189.
Dinka, 136, 147.
Dippil, 107.
Division, 19.
Dravidian, 104, 193.
Dual number, 75.
Duluth, 34.
Duodecimal, chap. v.
Dutch, 129.
Eaw, 24.
Ebon, 152.
Efik, 148, 185.
Encabellada, 22.
Encounter Bay, 108.
Ende, 68, 152.
English, 28, 38-44, 60, 81, 85, 89, 118, 123, 124, 129, 183, 200, 203, 208.
Eromanga, 96, 136, 154.
Eskimo, 16, 30, 31, 32, 36, 48, 51, 52, 54, 61, 64, 83, 137, 157, 159, 195,
196.
Essequibo, 166.
Europe (European), 27, 39, 168, 169, 179, 182, 183, 185, 204.
Eye, 14, 97.
Eyer's Sand Patch, 26.
Ewe, 64, 91.
Fall, 163.
Fate, 138, 155.
Fatuhiva, 130.
Feloop, 145.
Fernando Po, 150.
Fiji, 96.
Finger pantomime, 10, 23, 29, 67, 82.
Fingoe, 33.
Fist, 16, 59, 72.
Flinder's River, 24.
Flores, 68, 152.
Forefinger, 12, 15, 16, 54, 61, 91, 113.
Foulah, 147.
Fourth finger, 18.
Frazer's Island, 108.
French, 40, 41, 124, 129, 181, 182, 209.
Fuegan, 22.
Gaelic, 180.
Galibi, 138.
Gaul, 169, 182.
Georgia, 189.
German, 38-43, 129, 183.
Gesture, 18, 59.
Gola, 151.
Golo, 146.
Gonn Station, 110.
Goth, 169.
Greek, 86, 129, 168, 169.
Green Island, 45.
Greenland, 29, 52, 80, 158.
Guachi, 23, 31.
Guarani, 55, 138.
Guatemala, 205.
Guato, 142.
Guaycuru, 22.
Gudang, 24.
Haida, 112.
Hawaii, 113, 114, 116, 117.
Head, 71.
Heap, 8, 9, 25, 70, 77, 100.
Hebrew, 86, 89, 95.
Heiltsuk, 65, 88, 163.
Herero, 150.
Hervey Islands, 118.
Hidatsa, 80, 91.
Hill End, 109.
Himalaya, 193.
Hottentot, 80, 92.
Huasteca, 204.
Hudson's Bay, 48, 61.
Hun, 169.
Hunt, Leigh, 33.
Ibo, 185.
Icelandic, 129, 183.
Illinois, 91.
Index finger, 11, 14.
India, 96, 112, 131.
Indian, 8, 10, 13, 16, 17, 19, 32, 36, 54, 55, 59, 62, 65, 66, 79, 80, 82,
83, 89, 90, 98, 105, 112, 171, 201.
Indian Ocean, 63, 193.
Indo-European, 76.
Irish, 129, 180.
Italian, 39, 80, 124, 129, 203.
Jajowerong, 156.
Jallonkas, 146.
Jaloff, 146.
Japanese, 40, 86, 89, 93-95.
Java, 93, 120.
Jiviro, 61, 136.
Joints of fingers, 7, 18, 173.
Juri, 79.
Kamassin, 130.
Kamilaroi, 27, 107, 112.
Kamtschatka, 75, 157.
Kanuri, 136, 149.
Karankawa, 68.
Karen, 112.
Keppel Bay, 24.
Ki-Nyassa, 150.
Kiriri, 55, 138, 139, 167.
Kissi, 145.
Ki-Swahili, 42.
Ki-Yau, 150.
Klamath, 58, 59.
Knot, 7, 9, 19, 40, 93, 115.
Kolyma, 75.
Kootenay, 65.
Koriak, 75.
Kredy, 149.
Kru, 146.
Ku-Mbutti, 78.
Kunama, 151.
Kuri, 188.
Kusaie, 78, 80.
Kwakiutl, 45.
Labillardière, 85.
Labrador, 29.
Lake Kopperamana, 107.
Latin, 40, 44, 76, 81, 86, 124, 128, 168, 169, 181, 182.
Lazi, 189.
Left hand, 10-17, 54.
Leper's Island, 16.
Leptscha, 193.
Lifu, 143.
Little finger, 10-18, 48, 54, 61, 91.
Logone, 186.
London, 124.
Lower California, 29.
Luli, 118.
Lutuami, 164.
Maba, 80.
Macassar, 93.
Machine, Calculating, 19, 20.
Mackenzie River, 157.
Macuni, 174.
Madagascar, 8, 9.
Maipures, 15, 56.
Mairassis, 10.
Malagasy, 83, 95.
Malanta, 96.
Malay, 8, 45, 90, 93, 170.
Mallicolo, 152.
Manadu, 93.
Mandingo, 186.
Mangareva, 114.
Manx, 180.
Many, 2, 21-23, 25, 28, 100.
Maori, 64, 93, 122.
Marachowie, 26.
Maré, 84.
Maroura, 106.
Marquesas, 93, 114, 115.
Marshall Islands, 122, 152.
Massachusetts, 91, 159.
Mathematician, 2, 3, 35, 102, 127, 210.
Matibani, 151.
Matlaltzinca, 166.
Maya, 45, 46, 199, 205.
Mbayi, 111.
Mbocobi, 22.
Mbousha, 66.
Melanesia, 16, 22, 28, 84, 95.
Mende, 186.
Mexico, 29, 195, 201, 204, 208.
Miami, 91.
Micmac, 90, 160.
Middle finger, 12, 15, 62.
Mille, 122.
Minnal Yungar, 26.
Minsi, 162.
Mississaga, 44, 92.
Mississippi, 125.
Mocobi, 119.
Mohegan, 91.
Mohican, 172.
Mokko, 149.
Molele, 164.
Moneroo, 109.
Mongolian, 8.
Montagnais, 53, 54, 175.
Moree, 24.
Moreton Bay, 108.
Mort Noular, 107.
Mosquito, 69, 70, 121.
Mota, 95, 153.
Mpovi, 152.
Multiplication, 19, 33, 40, 43, 59.
Mundari, 193.
Mundo, 186.
Muralug, 17.
Murray River, 106, 109.
Muysca, 206.
Nagranda, 207.
Nahuatl, 141, 144, 177, 201, 205.
Nakuhiva, 116, 130.
Negro, 8, 9, 15, 29, 184.
Nengone, 63, 136.
New, 128-130.
New Caledonia, 154.
New Granada, 195.
New Guinea, 10, 152.
New Hebrides, 155.
New Ireland, 45.
New Zealand, 123.
Nez Perces, 65, 158.
Ngarrimowro, 110.
Niam Niam, 64, 136.
Nicaragua, 80.
Nicobar, 78, 193.
Nightingale, 4.
Nootka, 163, 198.
Norman River, 24.
North America, 28, 82, 171, 173, 176, 194, 201.
Notch, 7, 9, 93.
Numeral frame, 19.
Nupe, 149, 186.
Nusqually, 96.
Oceania, 115, 176.
Octonary, chap. v.
Odessa, 34.
Ojibwa, 84, 159.
Okanaken, 88.
Omaha, 161.
Omeo, 110.
Oregon, 58, 195.
Orejone, 23.
Orinoco, 54, 56, 195.
Ostrich, 71, 72.
Otomac, 15.
Otomi, 165, 199.
Ottawa, 159.
Oyster Bay, 79.
Pacific, 29, 113, 116, 117, 131.
Palm (of the hand), 12, 14, 15.
Palm Island, 156.
Pama, 136, 155.
Pampanaga, 66.
Papaa, 148.
Paraguay, 55, 71, 118, 195.
Parana, 119.
Paris, 182.
Pawnee, 91, 121, 162.
Pebble, 7-9, 19, 40, 93, 179.
Peno, 2.
Peru (Peruvian), 2, 22, 61, 206.
Philippine, 66.
Philology (Philologist), 128, 209, 210.
Phoenician, 90, 179.
Pigmy, 69, 70, 78.
Pikumbul, 57, 138.
Pines, Isle of, 153.
Pinjarra, 26.
Plenty, 25, 77.
Point Barrow, 30, 51, 64, 83, 137, 159.
Polynesia, 22, 28, 118, 130, 170.
Pondo, 33.
Popham Bay, 107.
Port Darwin, 109.
Port Essington, 24, 107.
Port Mackay, 26.
Port Macquarie, 109.
Puget Sound, 96.
Puri, 22, 92.
Quappa, 171, 172.
Quaternary, chap. v.
Queanbeyan, 24.
Quiche, 205.
Quichua, 61.
Rapid, 163.
Rarotonga, 114.
Richmond River, 109.
Right hand, 10-18, 54.
Right-handedness, 13, 14.
Ring finger, 15.
Rio Grande, 195.
Rio Napo, 22.
Rio Norte, 136, 199.
Russia (Russian), 30, 35.
Sahaptin, 158.
San Antonio, 136.
San Blas, 79, 80.
Sanskrit, 40, 92, 97, 128.
Sapibocone, 84, 167.
Saste (Shasta), 165.
Scratch, 7.
Scythian, 169.
Seed, 93.
Semitic, 89.
Senary, chap. v.
Sesake, 136, 155.
Several, 22.
Sexagesimal, 124, 208.
Shawnoe, 160.
Shell, 7, 19, 70, 93.
Shushwap, 88.
Siberia, 29, 30, 187, 190.
Sierra Leone, 83.
Sign language, 6.
Sioux, 83.
Slang, 124.
Slavonic, 40.
Snowy River, 110.
Soussou, 83, 147.
South Africa, 4, 15, 28.
South America, 2, 15, 22, 23, 27-29, 54, 57, 72, 76, 78, 79, 104, 110, 173,
174, 194, 201, 206.
Spanish, 2, 23, 42.
Splint, 7.
Stick, 7, 179.
Stlatlumh, 88.
Streaky Bay, 26.
String, 7, 9, 64, 71.
Strong's Island, 78.
Subtraction, 19, 44-47.
Sunda, 120.
Sweden (Swedish), 129, 132, 183.
Tacona, 2.
Taensa, 44.
Tagala, 66.
Tahiti, 114.
Tahuata, 115.
Tama, 111.
Tamanac, 54, 135.
Tambi, 120.
Tanna, 154.
Tarascan, 165.
Tariana, 174.
Tasmania, 24, 27, 79, 104, 106.
Tawgy, 130.
Tchetchnia, 188.
Tchiglit, 157, 196.
Tembu, 33.
Temne, 148.
Ternary, chap. v.
Terraba, 172.
Teutonic, 40, 41, 43, 179, 181, 208.
Texas, 69.
Thibet, 96.
Thumb, 10-18, 54, 59, 61, 62, 113, 173.
Thusch, 189.
Ticuna, 168.
Timukua, 165.
Tlingit, 136, 163, 197.
Tobi, 156.
Tonga, 33, 85.
Torres, 17, 96, 104, 105.
Totonaco, 205.
Towka, 78.
Triton's Bay, 152.
Tschukshi, 156, 191.
Tsimshian, 86, 164, 198.
Tweed River, 26.
Uainuma, 122.
Udi, 188.
Uea, 67, 153.
United States, 29, 83, 195.
Upper Yarra, 110.
Ureparapara, 153.
Vaturana, 96.
Vedda, 28, 31, 76, 174.
Vei, 16, 147, 185.
Victoria, 156.
Vilelo, 60.
Waiclatpu, 164.
Wales (Welsh), 35, 46, 141, 144, 177, 180.
Wallachia, 121.
Warrego, 107, 109.
Warrior Island, 107.
Wasp, 5.
Watchandie, 29, 77.
Watji, 120.
Weedookarry, 24.
Wimmera, 107.
Winnebago, 85.
Wiraduroi, 27, 108.
Wirri-Wirri, 108.
Wokke, 112.
Worcester, Mass., Schools of, 11.
Yahua, 168.
Yaruro, 139.
Yengen, 154.
Yit-tha, 109.
Yoruba, 33, 47, 64, 70, 185.
Yucatan, 195, 201.
Yuckaburra, 26.
Zamuco, 55, 60, 138, 139.
Zapara, 111.
Zulu, 16, 62.
Zuñi, 13, 14, 48, 49, 53, 54, 60, 83, 137.
FOOTNOTES:
[1] Brinton, D.G., _Essays of an Americanist_, p. 406; and _American Race_,
p. 359.
[2] This information I received from Dr. Brinton by letter.
[3] Tylor, _Primitive Culture_, Vol. I. p. 240.
[4] _Nature_, Vol. XXXIII. p. 45.
[5] Spix and Martius, _Travels in Brazil_, Tr. from German by H.E. Lloyd,
Vol. II. p. 255.
[6] De Flacourt, _Histoire de le grande Isle de Madagascar_, ch. xxviii.
Quoted by Peacock, _Encyc. Met._, Vol. I. p. 393.
[7] Bellamy, Elizabeth W., _Atlantic Monthly_, March, 1893, p. 317.
[8] _Grundriss der Sprachwissenschaft_, Bd. III. Abt. i., p. 94.
[9] Pruner-Bey, _Bulletin de la Société d'Anthr. de Paris_, 1861, p. 462.
[10] "Manual Concepts," _Am. Anthropologist_, 1892, p. 292.
[11] Tylor, _Primitive Culture_, Vol. I. p. 245.
[12] _Op. cit._, _loc. cit._
[13] "Aboriginal Inhabitants of Andaman Islands," _Journ. Anth. Inst._,
1882, p. 100.
[14] Morice, A., _Revue d'Anthropologie_, 1878, p. 634.
[15] Macdonald, J., "Manners, Customs, etc., of South African Tribes,"
_Journ. Anthr. Inst._, 1889, p. 290. About a dozen tribes are enumerated by
Mr. Macdonald: Pondos, Tembucs, Bacas, Tolas, etc.
[16] Codrington, R.H., _Melanesians, their Anthropology and Folk-Lore_, p.
353.
[17] _E.g._ the Zuñis. See Cushing's paper quoted above.
[18] Haddon, A.C., "Ethnography Western Tribes Torres Strait," _Journ.
Anth. Inst._, 1889, p. 305. For a similar method, see _Life in the Southern
Isles_, by W.W. Gill.
[19] Tylor, _Primitive Culture_, Vol. I. p. 246.
[20] Brinton, D.G., Letter of Sept. 23, 1893.
[21] _Ibid_. The reference for the Mbocobi, _infra_, is the same. See also
Brinton's _American Race_, p. 361.
[22] Tylor, _Primitive Culture_, Vol. I. p. 243.
[23] _Op. cit._, _loc. cit._
[24] Hyades, _Bulletin de la Société d'Anthr. de Paris_, 1887, p. 340.
[25] Wiener, C., _Pérou et Bolivie_, p. 360.
[26] Marcoy, P., _Travels in South America_, Vol. II p. 47. According to
the same authority, most of the tribes of the Upper Amazon cannot count
above 2 or 3 except by reduplication.
[27] _Op. cit._, Vol. II. p. 281.
[28] _Glossaria Linguarum Brasiliensium_. Bororos, p. 15; Guachi, p. 133;
Carajas, p. 265.
[29] Curr, E.M., _The Australian Race_, Vol. I. p. 282. The next eight
lists are, in order, from I. p. 294, III. p. 424, III. p. 114, III. p. 124,
II. p. 344, II. p. 308, I. p. 314, III. p. 314, respectively.
[30] Bonwick, J., _The Daily Life and Origin of the Tasmanians_, p. 144.
[31] Latham, _Comparative Philology_, p. 336.
[32] _The Australian Race_, Vol. I. p. 205.
[33] Mackenzie, A., "Native Australian Langs.," _Journ. Anthr. Inst._,
1874, p. 263.
[34] Curr, _The Australian Race_, Vol. II. p. 134. The next four lists are
from II. p. 4, I. p. 322, I. p. 346, and I. p. 398, respectively.
[35] Curr, _op. cit._, Vol. III. p. 50.
[36] _Op. cit._, Vol. III. p. 236.
[37] Müller, _Sprachwissenschaft_. II. i. p. 23.
[38] _Op. cit._, II. i. p. 31.
[39] Bonwick, _op. cit._, p. 143.
[40] Curr, _op. cit._, Vol. I. p. 31.
[41] Deschamps, _L'Anthropologie_, 1891, p. 318.
[42] Man, E.H. _Aboriginal Inhabitants of the Andaman Islands_, p. 32.
[43] Müller, _Sprachwissenschaft_, I. ii. p. 29.
[44] Oldfield, A., Tr. Eth. Soc. Vol. III. p. 291.
[45] Bancroft, H.H., _Native Races_, Vol. I. p. 564.
[46] "Notes on Counting, etc., among the Eskimos of Point Barrow." _Am.
Anthrop._, 1890, p. 38.
[47] _Second Voyage_, p. 556.
[48] _Personal Narrative_, Vol. I. p. 311.
[49] Burton, B.F., _Mem. Anthr. Soc. of London_, Vol. I. p. 314.
[50] _Confessions_. In collected works, Edinburgh, 1890, Vol. III. p. 337.
[51] Ellis, Robert, _On Numerals as Signs of Primeval Unity_. See also
_Peruvia Scythia_, by the same author.
[52] Stanley, H.M., _In Darkest Africa_, Vol. II. p. 493.
[53] Stanley, H.M., _Through the Dark Continent_, Vol. II. p. 486.
[54] Haumontè, Parisot, Adam, _Grammaire et Vocabulaire de la Langue
Taensa_, p. 20.
[55] Chamberlain, A.F., _Lang. of the Mississaga Indians of Skugog. Vocab._
[56] Boas, Fr., _Sixth Report on the Indians of the Northwest_, p. 105.
[57] Beauregard, O., _Bulletin de la Soc. d'Anthr. de Paris_, 1886, p. 526.
[58] Ray, S.H., _Journ. Anthr. Inst._, 1891, p. 8.
[59] _Op. cit._, p. 12.
[60] Müller, _Sprachwissenschaft_, IV. i. p. 136.
[61] Brinton, _The Maya Chronicles_, p. 50.
[62] Trumbull, _On Numerals in Am. Ind. Lang._, p. 35.
[63] Boas, Fr. This information was received directly from Dr. Boas. It has
never before been published.
[64] Bancroft, H.H., _Native Races_, Vol. II. p. 753. See also p. 199,
_infra_.
[65] Mann, A., "Notes on the Numeral Syst. of the Yoruba Nation," _Journ.
Anth. Inst._, 1886, p. 59, _et seq._
[66] Müller, _Sprachwissenschaft_, IV. i. p. 202.
[67] Trumbull, J.H., _On Numerals in Am. Ind. Langs._, p. 11.
[68] Cushing, F.H., "Manual Concepts," _Am. Anthr._, 1892, p. 289.
[69] Grimm, _Geschichte der deutschen Sprache_, Vol. I. p. 239.
[70] Murdoch, J., _American Anthropologist_, 1890, p. 39.
[71] Kleinschmidt, S., _Grammatik der Grönlandischen Sprache_, p. 37.
[72] Brinton, _The Arawak Lang. of Guiana_, p. 4.
[73] Petitot, E., _Dictionnaire de la langue Dènè-Dindjie_, p. lv.
[74] Gilij, F.S., _Saggio di Storia Am._, Vol. II. p. 333.
[75] Müller, _Sprachwissenschaft_, II. i. p. 389.
[76] _Op. cit._, p. 395.
[77] Müller, _Sprachwissenschaft_, II. i. p. 438.
[78] Peacock, "Arithmetic," in _Encyc. Metropolitana_, 1, p. 480.
[79] Brinton, D.G., "The Betoya Dialects," _Proc. Am. Philos. Soc._, 1892,
p. 273.
[80] Ridley, W., "Report on Australian Languages and Traditions." _Journ.
Anth. Inst._, 1873, p. 262.
[81] Gatschet, "Gram. Klamath Lang." _U.S. Geog. and Geol. Survey_, Vol.
II. part 1, pp. 524 and 536.
[82] Letter of Nov. 17, 1893.
[83] Müller, _Sprachwissenschaft_, II. i. p. 439.
[84] Hale, "Indians of No. West. Am.," _Tr. Am. Eth. Soc._, Vol. II. p. 82.
[85] Brinton, D.G., _Studies in So. Am. Native Languages_, p. 25.
[86] _Tr. Am. Philological Association_, 1874, p. 41.
[87] Tylor, _Primitive Culture_, Vol. I. p. 251.
[88] Müller, _Sprachwissenschaft_, IV. i. p. 27.
[89] See _infra_, Chapter VII.
[90] Ellis, A.B., _Ewe Speaking Peoples_, etc., p. 253.
[91] Tylor, _Primitive Culture_, Vol. I. p. 256.
[92] Stanley, _In Darkest Africa_, Vol. II. p. 493.
[93] Chamberlain, A.F., _Proc. Brit. Ass. Adv. of Sci._, 1892, p. 599.
[94] Boas, Fr., "Sixth Report on Northwestern Tribes of Canada," _Proc.
Brit. Ass. Adv. Sci._, 1890, p. 657.
[95] Hale, H., "Indians of Northwestern Am.," _Tr. Am. Eth. Soc._, Vol. II.
p. 88.
[96] _Op. cit._, p. 95.
[97] Müller, _Sprachwissenschaft_, II. ii. p. 147.
[98] Schoolcraft, _Archives of Aboriginal Knowledge_, Vol. IV. p. 429.
[99] Du Chaillu, P.B., _Tr. Eth. Soc._, London, Vol. I. p. 315.
[100] Latham, R.G., _Essays, chiefly Philological and Ethnographical_, p.
247. The above are so unlike anything else in the world, that they are not
to be accepted without careful verification.
[101] Pott, _Zählmethode_, p. 45.
[102] Gatschet, A.S., _The Karankawa Indians, the Coast People of Texas_.
The meanings of 6, 7, 8, and 9 are conjectural with me.
[103] Stanley, H.M., _In Darkest Africa_, Vol. II. p. 492.
[104] Müller, _Sprachwissenschaft_, II. i. p. 317.
[105] Toy, C.H., _Trans. Am. Phil. Assn._, 1878, p. 29.
[106] Burton, R.F., _Mem. Anthrop. Soc. of London_. 1, p. 314. In the
illustration which follows, Burton gives 6820, instead of 4820; which is
obviously a misprint.
[107] Dobrizhoffer, _History of the Abipones_, Vol. II. p. 169.
[108] Sayce, A.H., _Comparative Philology_, p. 254.
[109] _Tr. Eth. Society of London _, Vol. III. p. 291.
[110] Ray, S.H., _Journ. Anthr. Inst._, 1889, p. 501.
[111] Stanley, _In Darkest Africa_, Vol. II. p. 492.
[112] _Op. cit._, _loc. cit._
[113] Tylor, _Primitive Culture_, Vol. I. p. 249.
[114] Müller, _Sprachwissenschaft_, IV. i. p. 36.
[115] Martius, _Glos. Ling. Brasil._, p. 271.
[116] Tylor, _Primitive Culture_, Vol. I. p. 248.
[117] Roth, H. Ling, _Aborigines of Tasmania_, p. 146.
[118] Lull, E.P., _Tr. Am. Phil, Soc._, 1873, p. 108.
[119] Ray, S.H. "Sketch of Api Gram.," _Journ. Anthr. Inst._, 1888, p. 300.
[120] Kleinschmidt, S., _Grammatik der Grönlandischen Spr._, p. 39.
[121] Müller, _Sprachwissenschaft_, I. ii. p. 184.
[122] _Op. cit._, I. ii. p. 18, and II. i. p. 222.
[123] Squier, G.E., _Nicaragua_, Vol. II. p. 326.
[124] Schoolcraft, H.R., _Archives of Aboriginal Knowledge_, Vol. II. p.
208.
[125] Tylor, _Primitive Culture_, Vol. I. p. 264.
[126] Goedel, "Ethnol. des Soussous," _Bull. de la Soc. d'Anthr. de Paris_,
1892, p. 185.
[127] Ellis, W., _History of Madagascar_, Vol. I. p. 507.
[128] Beauregard, O., _Bull. de la Soc. d'Anthr. de Paris_, 1886, p. 236.
[129] Schoolcraft, H.R., _Archives of Aboriginal Knowledge_, Vol. II. p.
207.
[130] Tylor, _Primitive Culture_, Vol. I. p. 249.
[131] _Op. cit._ Vol. I. p. 250.
[132] Peacock, _Encyc. Metropolitana_, 1, p. 478.
[133] _Op. cit._, _loc. cit._
[134] Schoolcraft, H.R., _Archives of Aboriginal Knowledge_, Vol. II. p.
213.
[135] _Op. cit._, p. 216.
[136] _Op. cit._, p. 206.
[137] Mariner, _Gram. Tonga Lang._, last part of book. [Not paged.]
[138] Morice, A.G., "The Déné Langs," _Trans. Can. Inst._, March 1890, p.
186.
[139] Boas, Fr., "Fifth Report on the Northwestern Tribes of Canada,"
_Proc. Brit. Ass. Adv. of Science_, 1889, p. 881.
[140] _Do. Sixth Rep._, 1890, pp. 684, 686, 687.
[141] _Op. cit._, p. 658.
[142] Bancroft, H.H., _Native Races_, Vol. II. p. 499.
[143] _Tr. Ethnological Soc. of London_, Vol. IV. p. 92.
[144] Any Hebrew lexicon.
[145] Schröder, P., _Die Phönizische Sprache, _p. 184 _et seq._
[146] Müller, _Sprachwissenschaft_, II. ii. p. 147.
[147] _On Numerals in Am. Indian Languages._
[148] Ellis, A.B., _Ewe Speaking Peoples_, etc., p. 253. The meanings here
given are partly conjectural.
[149] Pott, _Zählmethode_, p. 29.
[150] Schoolcraft, _op. cit._, Vol. IV. p. 429.
[151] Trumbull, _op. cit._
[152] Chamberlain, A.F., _Lang, of the Mississaga Indians_, Vocab.
[153] Crawfurd, _Hist. Ind. Archipelago_, 1, p. 258.
[154] Hale, H., _Eth. and Philol._, Vol. VII.; Wilkes, _Expl. Expedition_,
Phil. 1846, p. 172.
[155] Crawfurd, _op. cit._, 1, p. 258.
[156] _Op. cit._, _loc. cit._
[157] Bancroft, H.H., _Native Races_, Vol. II. p. 498.
[158] Vignoli, T., _Myth and Science_, p. 203.
[159] Codrington, R.H., _The Melanesian Languages_, p. 249.
[160] _Op. cit._, _loc. cit._
[161] Codrington, R.H., _The Melanesian Languages_, p. 249.
[162] Wickersham, J., "Japanese Art on Puget Sound," _Am. Antiq._, 1894, p.
79.
[163] Codrington, R.H., _op. cit._, p. 250.
[164] Tylor, _Primitive Culture_, Vol. I. p. 252.
[165] Compare a similar table by Chase, _Proc. Amer. Philos. Soc._, 1865,
p. 23.
[166] _Leibnitzii Opera_, III. p. 346.
[167] Pruner-Bey, _Bulletin de la Soc. d'Anthr. de Paris_, 1860, p. 486.
[168] Curr, E.M., _The Australian Race_, Vol. I. p. 32.
[169] Haddon, A.C., "Western Tribes of the Torres Straits," _Journ. Anthr.
Inst._, 1889, p. 303.
[170] Taplin, Rev. G., "Notes on a Table of Australian Languages," _Journ.
Anthr. Inst.,_ 1872, p. 88. The first nine scales are taken from this
source.
[171] Latham, R.G., _Comparative Philology_, p. 352.
[172] It will be observed that this list differs slightly from that given
in Chapter II.
[173] Curr, E.M., _The Australian Race_, Vol. III. p. 684.
[174] Bonwick, _Tasmania_, p. 143.
[175] Lang, J.D., _Queensland_, p. 435.
[176] Bonwick, _Tasmania_, p. 143.
[177] Müller, _Sprachwissenschaft_, II. i. p. 58.
[178] _Op. cit._, II. i. p. 70.
[179] _Op. cit._, II. i. p. 23.
[180] Barlow, H., "Aboriginal Dialects of Queensland," _Journ. Anth.
Inst._, 1873, p. 171.
[181] Curr, E.M., _The Australian Race_, Vol. II. p. 26.
[182] _Op. cit._, Vol. II. p. 208.
[183] _Op. cit._, Vol. II. p. 278.
[184] _Op. cit._, Vol. II. p. 288.
[185] _Op. cit._, Vol. I. p. 258.
[186] _Op. cit._, Vol. I. p. 316.
[187] _Op. cit._, Vol. III. p. 32. The next ten lists are taken from the
same volume, pp. 282, 288, 340, 376, 432, 506, 530, 558, 560, 588,
respectively.
[188] Brinton, _The American Race_, p. 351.
[189] Martius, _Glossaria Ling. Brazil._, p. 307.
[190] _Op. cit._, p. 148.
[191] Müller, _Sprachwissenschaft_, II. i. p. 438.
[192] Peacock, "Arithmetic," _Encyc. Metropolitana_, 1, p. 480.
[193] Brinton, _Studies in So. Am. Native Langs._, p. 67.
[194] _Op. cit._, _loc. cit._
[195] Brinton, _Studies in So. Am. Native Langs._, p. 67. The meanings of
the numerals are from Peacock, _Encyc. Metropolitana_, 1, p. 480.
[196] Mason, _Journ. As. Soc. of Bengal_, Vol. XXVI. p. 146.
[197] Curr, E.M., _The Australian Race_, Vol. III. p. 108.
[198] Bancroft, H.H., _Native Races_, Vol. I. p. 274.
[199] Clarke, Hyde, _Journ. Anthr. Inst._, 1872, p. clvii. In the article
from which this is quoted, no evidence is given to substantiate the
assertion made. It is to be received with great caution.
[200] Hale, H., _Wilkes Exploring Expedition_, Vol. VII. p. 172.
[201] _Op. cit._, p. 248.
[202] Hale, _Ethnography and Philology, _p. 247.
[203] _Loc. cit._
[204] Ellis, _Polynesian Researches_, Vol. IV. p. 341.
[205] Gill, W.W., _Myths and Songs of the South Pacific_, p. 325.
[206] Peacock, "Arithmetic," _Encyc. Metropolitana_, 1, p. 479.
[207] Peacock, _Encyc. Metropolitana_, 1, p. 480.
[208] _Sprachverschiedenheit_, p. 30.
[209] Crawfurd, _History of the Indian Archipelago_, Vol. I. p. 256.
[210] Pott, _Zählmethode_, p. 39.
[211] _Op. cit._, p. 41.
[212] Müller, _Sprachwissenschaft_, II. i. p. 317. See also Chap. III.,
_supra_.
[213] Long, S.H., _Expedition_, Vol. II. p. lxxviii.
[214] Martius, _Glossaria Ling. Brasil._, p. 246.
[215] Hale, _Ethnography and Philology_, p. 434.
[216] Müller, _Sprachwissenschaft_, II. ii. p. 82.
[217] The information upon which the above statements are based was
obtained from Mr. W.L. Williams, of Gisborne, N.Z.
[218] _Primitive Culture_, Vol. I. p. 268.
[219] Ralph, Julian, _Harper's Monthly_, Vol. 86, p. 184.
[220] Lappenberg, J.M., _History of Eng. under the Anglo-Saxon Kings_, Vol.
I. p. 82.
[221] The compilation of this table was suggested by a comparison found in
the _Bulletin Soc. Anth. de Paris_, 1886, p. 90.
[222] Hale, _Ethnography and Philology_, p. 126.
[223] Müller, _Sprachwissenschaft_, II. ii. p. 183.
[224] Bachofen, J.J., _Antiquarische Briefe_, Vol. I. pp. 101-115, and Vol.
II. pp. 1-90.
[225] An extended table of this kind may be found in the last part of
Nystrom's _Mechanics_.
[226] Schubert, H., quoting Robert Flegel, in Neumayer's _Anleitung zu
Wissenschaftlichen Beobachtung auf Reisen_, Vol. II. p. 290.
[227] These numerals, and those in all the sets immediately following,
except those for which the authority is given, are to be found in Chapter
III.
[228] Codrington, _The Melanesian Languages_, p. 222.
[229] Müller, _Sprachwissenschaft_, II. ii. p. 83.
[230] _Op. cit._, I. ii. p. 55. The next two are the same, p. 83 and p.
210. The meaning given for the Bari _puök_ is wholly conjectural.
[231] Gallatin, "Semi-civilized Nations," _Tr. Am. Eth. Soc._, Vol. I. p.
114.
[232] Müller, _Sprachwissenschaft_, II. ii. p. 80. Erromango, the same.
[233] Boas, Fr., _Proc. Brit. Ass'n. Adv. Science_, 1889, p. 857.
[234] Hankel, H., _Geschichte der Mathematik_, p. 20.
[235] Murdoch, J., "Eskimos of Point Barrow," _Am. Anthr._, 1890, p. 40.
[236] Martius, _Glos. Ling. Brasil._, p. 360.
[237] Du Graty, A.M., _La République du Paraguay_, p. 217.
[238] Codrington, _The Melanesian Languages_, p. 221.
[239] Müller, _Sprachwissenschaft_, II. i. p. 363.
[240] Spurrell, W., _Welsh Grammar_, p. 59.
[241] Olmos, André de, _Grammaire Nahuatl ou Mexicaine_, p. 191.
[242] Moncelon, _Bull. Soc. d'Anthr. de Paris_, 1885, p. 354. This is a
purely digital scale, but unfortunately M. Moncelon does not give the
meanings of any of the numerals except the last.
[243] Ellis, _Peruvia Scythia_, p. 37. Part of these numerals are from
Martius, _Glos. Brasil._, p. 210.
[244] Codrington, _The Melanesian Languages_, p. 236.
[245] Schweinfurth, G., _Linguistische Ergebnisse einer Reise nach
Centralafrika_, p. 25.
[246] Park, M., _Travels in the Interior Districts of Africa_, p. 8.
[247] Pott, _Zählmethode_, p. 37.
[248] _Op. cit._, p. 39.
[249] Müller, _Sprachwissenschaft_, IV. i. p. 101. The Kru scale, kindred
with the Basa, is from the same page.
[250] Park, in Pinkerton's _Voyages and Travels_, Vol. XVI. p. 902.
[251] Park, _Travels_, Vol. I. p. 16.
[252] Schweinfurth, G., _Linguistische Ergebnisse einer Reise nach
Centralafrika_, p. 78.
[253] Park, _Travels_, Vol. I. p. 58.
[254] Goedel, "Ethnol. des Soussous," _Bull. Soc. Anth. Paris_, 1892, p.
185.
[255] Müller, _Sprachwissenschaft_, I. ii. p. 114. The Temne scale is from
the same page. These two languages are closely related.
[256] _Op. cit._, I. ii. p. 155.
[257] _Op. cit._, I. ii. p. 55.
[258] Long, C.C., _Central Africa_, p. 330.
[259] Müller, _Sprachwissenschaft_, IV. i. p. 105.
[260] Pott, _Zählmethode_, p. 41.
[261] Müller, _op. cit._, I. ii. p. 140.
[262] Müller, _Sprachwissenschaft_, IV. i. p. 81.
[263] Pott, _Zählmethode_, p. 41.
[264] Müller, _op. cit._, I. ii., p. 210.
[265] Pott, _Zählmethode_, p. 42.
[266] Schweinfurth, _Linguistische Ergebnisse_, p. 59.
[267] Müller, _Sprachwissenschaft_, I. ii. p. 261. The "ten" is not given.
[268] Stanley, _Through the Dark Continent_, Vol. II. p. 490. Ki-Nyassa,
the same page.
[269] Müller, _op. cit._, I. ii. p. 261.
[270] Du Chaillu, _Adventures in Equatorial Africa_, p. 534.
[271] Müller, _Sprachwissenschaft_, III. i. p. 65.
[272] Du Chaillu, _Adventures in Equatorial Africa_, p. 533.
[273] Müller, _op. cit._, III. ii. p. 77.
[274] Balbi, A., _L'Atlas Eth._, Vol. I. p. 226. In Balbi's text 7 and 8
are ansposed. _Taru_ for 5 is probably a misprint for _tana_.
[275] Du Chaillu, _op. cit._, p. 533. The next scale is _op. cit._, p. 534.
[276] Beauregard, O., _Bull. Soc. Anth. de Paris_, 1886, p. 526.
[277] Pott, _Zählmethode_, p. 46.
[278] _Op. cit._, p. 48.
[279] Turner, _Nineteen Years in Polynesia_, p. 536.
[280] Erskine, J.E., _Islands of the Western Pacific_, p. 341.
[281] _Op. cit._, p. 400.
[282] Codrington, _Melanesian Languages_, pp. 235, 236.
[283] Peacock, _Encyc. Met._, Vol. 1. p. 385. Peacock does not specify the
dialect.
[284] Erskine, _Islands of the Western Pacific_, p. 360.
[285] Turner, G., _Samoa a Hundred Years Ago_, p. 373. The next three
scales are from the same page of this work.
[286] Codrington, _Melanesian Languages_, p. 235. The next four scales are
from the same page. Perhaps the meanings of the words for 6 to 9 are more
properly "more 1," "more 2," etc. Codrington merely indicates their
significations in a general way.
[287] Hale, _Ethnography and Philology_, p. 429. The meanings of 6 to 9 in
this and the preceding are my conjectures.
[288] Müller, _Sprachwissenschaft_, IV. i. p. 124.
[289] Aymonier, E., _Dictionnaire Francaise-Cambodgien_.
[290] Müller, _Op. cit._, II. i. p. 139.
[291] Müller, _Sprachwissenschaft_, II. i. p. 123.
[292] Wells, E.R., Jr., and John W. Kelly, Bureau of Ed., Circ. of Inf.,
No. 2, 1890.
[293] Pott, _Zählmethode_, p. 57.
[294] Müller, _Op. cit._, II. i. p. 161.
[295] Petitot, _Vocabulaire Française Esquimau_, p. lv.
[296] Müller, _Sprachwissenschaft_, II. i. p. 253.
[297] Müller, _Op. cit._, II. I. p. 179, and Kleinschmidt, _Grönlandisches
Grammatik_.
[298] Adam, L., _Congres Int. des Am._, 1877, p. 244 (see p. 162 _infra_).
[299] Gallatin, "Synopsis of Indian Tribes," _Trans. Am. Antq. Soc._, 1836,
p. 358. The next fourteen lists are, with the exception of the Micmac, from
the same collection. The meanings are largely from Trumbull, _op. cit._
[300] Schoolcraft, _Archives of Aboriginal Knowledge_, Vol. II. p. 211.
[301] Schoolcraft, _Archives of Aboriginal Knowledge_, Vol. V. p. 587.
[302] In the Dakota dialects 10 is expressed, as here, by a word signifying
that the fingers, which have been bent down in counting, are now
straightened out.
[303] Boas, _Fifth Report B.A.A.S._, 1889. Reprint, p. 61.
[304] Boas, _Sixth Report B.A.A.S._, 1890. Reprint, p. 117. Dr. Boas does
not give the meanings assigned to 7 and 8, but merely states that they are
derived from 2 and 3.
[305] _Op. cit._, p. 117. The derivations for 6 and 7 are obvious, but the
meanings are conjectural.
[306] Boas, _Sixth Report B.A.A.S._, 1889. Reprint, pp. 158, 160. The
meanings assigned to the Tsimshian 8 and to Bilqula 6 to 8 are conjectural.
[307] Hale, _Ethnography and Philology_, p. 619.
[308] _Op. cit._, _loc. cit._
[309] Hale, _Ethnography and Philology_, p. 619.
[310] Müller, _Sprachwissenschaft_, II. i. p. 436.
[311] _Op. cit._, IV. i. p. 167.
[312] _Op. cit._, II. i. p. 282.
[313] _Op. cit._, II. i. p. 287. The meanings given for the words for 7, 8,
9 are conjectures of my own.
[314] Müller, _Sprachwissenschaft_, II. i. p. 297.
[315] Pott, _Zählmethode_, p. 90.
[316] Müller, _op. cit._, II. i. p. 379.
[317] Gallatin, "Semi-Civilized Nations of Mexico and Central America,"
_Tr. Am. Ethn. Soc._, Vol. I. p. 114.
[318] Adam, Lucien, _Congres Internationale des Americanistes_, 1877, Vol.
II. p. 244.
[319] Müller, _Sprachwissenschaft_, II. i. p. 395. I can only guess at the
meanings of 6 to 9. They are obviously circumlocutions for 5-1, 5-2, etc.
[320] _Op. cit._, p. 438. Müller has transposed these two scales. See
Brinton's _Am. Race_, p. 358.
[321] Marcoy, P., _Tour du Monde_, 1866, 2ème sem. p. 148.
[322] _Op. cit._, p. 132. The meanings are my own conjectures.
[323] An elaborate argument in support of this theory is to be found in
Hervas' celebrated work, _Arithmetica di quasi tutte le nazioni
conosciute_.
[324] See especially the lists of Hale, Gallatin, Trumbull, and Boas, to
which references have been given above.
[325] Thiel, B.A., "Vocab. der Indianier in Costa Rica," _Archiv für
Anth._, xvi. p. 620.
[326] These three examples are from A.R. Wallace's _Narrative of Travels on
the Amazon and Rio Negro_, vocab. Similar illustrations may be found in
Martius' _Glos. Brasil_.
[327] Martius, _Glos. Brasil._, p. 176.
[328] Adam, L., _Congres International des Americanistes_, 1877, Vol. II.
p. 244. Given also _supra_, p. 53.
[329] O'Donovan, _Irish Grammar_, p. 123.
[330] Armstrong, R.A., _Gaelic Dict._, p. xxi.
[331] Spurrell, _Welsh Dictionary_.
[332] Kelly, _Triglot Dict._, pub. by the Manx Society.
[333] Guillome, J., _Grammaire Française-Bretonne_, p. 27.
[334] Gröber, G., _Grundriss der Romanischen Philologie_, Bd. I. p. 309.
[335] Pott, _Zählmethode_, p. 88.
[336] Van Eys, _Basque Grammar_, p. 27.
[337] Pott, _Zählmethode_, p. 101.
[338] _Op. cit._, p. 78.
[339] Müller, _Sprachwissenschaft_, I. ii. p. 124.
[340] _Op. cit._, p. 155.
[341] _Op. cit._, p. 140.
[342] _Op. cit._, _loc. cit._
[343] Schweinfurth, _Reise nach Centralafrika_, p. 25.
[344] Müller, _Sprachwissenschaft_, IV. i. p. 83.
[345] _Op. cit._, IV. i. p. 81.
[346] _Op. cit._, I. ii. p. 166.
[347] Long, C.C., _Central Africa_, p. 330.
[348] Peacock, _Encyc. Met._, Vol. I. p. 388.
[349] Müller, _Sprachwissenschaft_, III. ii. p. 64. The next seven scales
are from _op. cit._, pp. 80, 137, 155, 182, 213.
[350] Pott, _Zählmethode_, p. 83.
[351] _Op. cit._, p. 83,--Akari, p. 84; Circassia, p. 85.
[352] Müller, _Sprachwissenschaft_, II. i. p. 140.
[353] Pott, _Zählmethode_, p. 87.
[354] Müller, _Sprachwissenschaft_, II. ii. p. 346.
[355] _Op. cit._, III. i. p. 130.
[356] Man, E.H., "Brief Account of the Nicobar Islands," _Journ. Anthr.
Inst._, 1885, p. 435.
[357] Wells, E.R., Jr., and Kelly, J.W., "Eng. Esk. and Esk. Eng. Vocab.,"
Bureau of Education Circular of Information, No. 2, 1890, p. 65.
[358] Petitot, E., _Vocabulaire Française Esquimau_, p. lv.
[359] Boas, Fr., _Proc. Brit. Ass. Adv. Sci._, 1889, p. 857.
[360] Boas, _Sixth Report on the Northwestern Tribes of Canada_, p. 117.
[361] Boas, Fr., _Fifth Report on the Northwestern Tribes of Canada_, p.
85.
[362] Gallatin, _Semi-Civilized Nations_, p. 114. References for the next
two are the same.
[363] Bancroft, H.H., _Native Races of the Pacific States_, Vol. II. p.
763. The meanings are from Brinton's _Maya Chronicles_, p. 38 _et seq._
[364] Brinton, _Maya Chronicles_, p. 44.
[365] Siméon Rémi, _Dictionnaire de la langue nahuatl_, p. xxxii.
[366] An error occurs on p. xxxiv of the work from which these numerals are
taken, which makes the number in question appear as 279,999,999 instead of
1,279,999,999.
[367] Gallatin, "Semi-Civilized Nations of Mexico and Central America,"
_Tr. Am. Ethn. Soc._ Vol. I. p. 114.
[368] Pott, _Zählmethode_, p. 89. The Totonacos were the first race Cortez
encountered after landing in Mexico.
[369] _Op. cit._, p. 90. The Coras are of the Mexican state of Sonora.
[370] Gallatin, _Semi-Civilized Nations_, p. 114.
[371] Humboldt, _Recherches_, Vol. II. p. 112.
[372] Squier, _Nicaragua_, Vol. II. p. 326.
[373] Gallatin, _Semi-Civilized Nations_, p. 57.
End of Project Gutenberg's The Number Concept, by Levi Leonard Conant