The Number Concept - Its Origin and Development
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CHAPTER VI.

THE QUINARY SYSTEM.

The origin of the quinary mode of counting has been discussed with some fulness in a preceding chapter, and upon that question but little more need be said. It is the first of the natural systems. When the savage has finished his count of the fingers of a single hand, he has reached this natural number base. At this point he ceases to use simple numbers, and begins the process of compounding. By some one of the numerous methods illustrated in earlier chapters, he passes from 5 to 10, using here the fingers of his second hand. He now has two fives; and, just as we say "twenty," i.e. two tens, he says "two hands," "the second hand finished," "all the fingers," "the fingers of both hands," "all the fingers come to an end," or, much more rarely, "one man." That is, he is, in one of the many ways at his command, saying "two fives." At 15 he has "three hands" or "one foot"; and at 20 he pauses with "four hands," "hands and feet," "both feet," "all the fingers of hands and feet," "hands and feet finished," or, more probably, "one man." All these modes of expression are strictly natural, and all have been found in the number scales which were, and in many cases still are, in daily use among the uncivilized races of mankind.

In its structure the quinary is the simplest, the most primitive, of the natural systems. Its base is almost always expressed by a word meaning "hand," or by some equivalent circumlocution, and its digital origin is usually traced without difficulty. A consistent formation would require the expression of 10 by some phrase meaning "two fives," 15 by "three fives," etc. Such a scale is the one obtained from the Betoya language, already mentioned in Chapter III., where the formation of the numerals is purely quinary, as the following indicate:[227]

5. teente = 1 hand. 10. cayaente, or caya huena = 2 hands. 15. toazumba-ente = 3 hands. 20. caesa-ente = 4 hands.

The same formation appears, with greater or less distinctness, in many of the quinary scales already quoted, and in many more of which mention might be made. Collecting the significant numerals from a few such scales, and tabulating them for the sake of convenience of comparison, we see this point clearly illustrated by the following:

TAMANAC.

5. amnaitone = 1 hand. 10. amna atse ponare = 2 hands.

ARAWAK, GUIANA.

5. abba tekkabe = 1 hand. 10. biamantekkabe = 2 hands.

JIVIRO.

NIAM NIAM

5. biswe 10. bauwe = 2d 5.

NENGONES

5. se dono = the end (of the fingers of 1 hand). 10. rewe tubenine = 2 series (of fingers).

SESAKE.[228]

5. lima = hand. 10. dua lima = 2 hands.

AMBRYM.[229]

5. lim = hand. 10. ra-lim = 2 hands.

PAMA.[229]

5. e-lime = hand. 10. ha-lua-lim = the 2 hands.

DINKA.[230]

5. wdyets. 10. wtyer, or wtyar = 5 x 2.

BARI

5. kanat 10. puoek = 5 + 5?

KANURI

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

RIO NORTE AND SAN ANTONIO.[231]

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

API.[232]

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

ERROMANGO

5. suku-rim. 10. nduru-lim = 2 x 5.

TLINGIT, BRITISH COLUMBIA.[233]

5. kedjin (from djin = hand). 10. djinkat = both hands?

Thus far the quinary formation is simple and regular; and in view of the evidence with which these and similar illustrations furnish us, it is most surprising to find an eminent authority making the unequivocal statement that the number 10 is nowhere expressed by 2 fives[234]—that all tribes which begin their count on a quinary base express 10 by a simple word. It is a fact, as will be fully illustrated in the following pages, that quinary number systems, when extended, usually merge into either the decimal or the vigesimal. The result is, of course, a compound of two, and sometimes of three, systems in one scale. A pure quinary or vigesimal number system is exceedingly rare; but quinary scales certainly do exist in which, as far as we possess the numerals, no trace of any other influence appears. It is also to be noticed that some tribes, like the Eskimos of Point Barrow, though their systems may properly be classed as mixed systems, exhibit a decided preference for 5 as a base, and in counting objects, divided into groups of 5, obtaining the sum in this way.[235]

But the savage, after counting up to 10, often finds himself unconsciously impelled to depart from his strict reckoning by fives, and to assume a new basis of reference. Take, for example, the Zuni system, in which the first 2 fives are:

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

It will be noticed that the Zuni does not say "two hands," or "the fingers of both hands," but simply "all the fingers." The 5 is no longer prominent, but instead the mere notion of one entire count of the fingers has taken its place. The division of the fingers into two sets of five each is still in his mind, but it is no longer the leading idea. As the count proceeds further, the quinary base may be retained, or it may be supplanted by a decimal or a vigesimal base. How readily the one or the other may predominate is seen by a glance at the following numerals:

GALIBI.[236]

5. atoneigne oietonai = 1 hand. 10. oia batoue = the other hand. 20. poupoupatoret oupoume = feet and hands. 40. opoupoume = twice the feet and hands.

GUARANI.[237]

5. ace popetei = 1 hand. 10. ace pomocoi = 2 hands. 20. acepo acepiabe = hands and feet.

FATE.[238]

5. lima = hand. 10. relima = 2 hands. 20. relima rua = (2 x 5) x 2.

KIRIRI

5. mibika misa = 1 hand. 10. mikriba misa sai = both hands. 20. mikriba nusa ideko ibi sai = both hands together with the feet.

ZAMUCO

5. tsuena yimana-ite = ended 1 hand. 10. tsuena yimana-die = ended both hands. 20. tsuena yiri-die = ended both feet.

PIKUMBUL

5. mulanbu. 10. bularin murra = belonging to the two hands. 15. mulanba dinna = 5 toes added on (to the 10 fingers). 20. bularin dinna = belonging to the 2 feet.

YARUROS.[239]

5. kani-iktsi-mo = 1 hand alone. 10. yowa-iktsi-bo = all the hands. 15. kani-tao-mo = 1 foot alone. 20. kani-pume = 1 man.

By the time 20 is reached the savage has probably allowed his conception of any aggregate to be so far modified that this number does not present itself to his mind as 4 fives. It may find expression in some phraseology such as the Kiriris employ—"both hands together with the feet"—or in the shorter "ended both feet" of the Zamucos, in which case we may presume that he is conscious that his count has been completed by means of the four sets of fives which are furnished by his hands and feet. But it is at least equally probable that he instinctively divides his total into 2 tens, and thus passes unconsciously from the quinary into the decimal scale. Again, the summing up of the 10 fingers and 10 toes often results in the concept of a single whole, a lump sum, so to speak, and the savage then says "one man," or something that gives utterance to this thought of a new unit. This leads the quinary into the vigesimal scale, and produces the combination so often found in certain parts of the world. Thus the inevitable tendency of any number system of quinary origin is toward the establishment of another and larger base, and the formation of a number system in which both are used. Wherever this is done, the greater of the two bases is always to be regarded as the principal number base of the language, and the 5 as entirely subordinate to it. It is hardly correct to say that, as a number system is extended, the quinary element disappears and gives place to the decimal or vigesimal, but rather that it becomes a factor of quite secondary importance in the development of the scale. If, for example, 8 is expressed by 5-3 in a quinary decimal system, 98 will be 9 x 10 + 5-3. The quinary element does not disappear, but merely sinks into a relatively unimportant position.

One of the purest examples of quinary numeration is that furnished by the Betoya scale, already given in full in Chapter III., and briefly mentioned at the beginning of this chapter. In the simplicity and regularity of its construction it is so noteworthy that it is worth repeating, as the first of the long list of quinary systems given in the following pages. No further comment is needed on it than that already made in connection with its digital significance. As far as given by Dr. Brinton the scale is:

1. tey. 2. cayapa. 3. toazumba. 4. cajezea = 2 with plural termination. 5. teente = hand. 6. teyente tey = hand 1. 7. teyente cayapa = hand 2. 8. teyente toazumba = hand 3. 9. teyente caesea = hand 4. 10. caya ente, or caya huena = 2 hands. 11. caya ente-tey = 2 hands 1. 15. toazumba-ente = 3 hands. 16. toazumba-ente-tey = 3 hands 1. 20. caesea ente = 4 hands.

A far more common method of progression is furnished by languages which interrupt the quinary formation at 10, and express that number by a single word. Any scale in which this takes place can, from this point onward, be quinary only in the subordinate sense to which allusion has just been made. Examples of this are furnished in a more or less perfect manner by nearly all so-called quinary-vigesimal and quinary-decimal scales. As fairly representing this phase of number-system structure, I have selected the first 20 numerals from the following languages:

WELSH.[240]

1. un. 2. dau. 3. tri. 4. pedwar. 5. pump. 6. chwech. 7. saith. 8. wyth. 9. naw. 10. deg. 11. un ar ddeg = 1 + 10. 12. deuddeg = 2 + 10. 13. tri ar ddeg = 3 + 10. 14. pedwar ar ddeg = 4 + 10. 15. pymtheg = 5 + 10. 16. un ar bymtheg = 1 + 5 + 10. 17. dau ar bymtheg = 2 + 5 + 10. 18. tri ar bymtheg = 3 + 5 + 10. 19. pedwar ar bymtheg = 4 + 5 + 10. 20. ugain.

NAHUATL.[241]

1. ce. 2. ome. 3. yei. 4. naui. 5. macuilli. 6. chiquacen = [5] + 1. 7. chicome = [5] + 2. 8. chicuey = [5] + 3. 9. chiucnaui = [5] + 4. 10. matlactli. 11. matlactli oce = 10 + 1. 12. matlactli omome = 10 + 2. 13. matlactli omey = 10 + 3. 14. matlactli onnaui = 10 + 4. 15. caxtolli. 16. caxtolli oce = 15 + 1. 17. caxtolli omome = 15 + 2. 18. caxtolli omey = 15 + 3. 19. caxtolli onnaui = 15 + 4. 20. cempualli = 1 account.

CANAQUE[242] NEW CALEDONIA.

1. chaguin. 2. carou. 3. careri. 4. caboue 5. cani. 6. cani-mon-chaguin = 5 + 1. 7. cani-mon-carou = 5 + 2. 8. cani-mon-careri = 5 + 3. 9. cani-mon-caboue = 5 + 4. 10. panrere. 11. panrere-mon-chaguin = 10 + 1. 12. panrere-mon-carou = 10 + 2. 13. panrere-mon-careri = 10 + 3. 14. panrere-mon-caboue = 10 + 4. 15. panrere-mon-cani = 10 + 5. 16. panrere-mon-cani-mon-chaguin = 10 + 5 + 1. 17. panrere-mon-cani-mon-carou = 10 + 5 + 2. 18. panrere-mon-cani-mon-careri = 10 + 5 + 3. 19. panrere-mon-cani-mon-caboue = 10 + 5 + 4. 20. jaquemo = 1 person.

GUATO.[243]

1. cenai. 2. dououni. 3. coum. 4. dekai. 5. quinoui. 6. cenai-caicaira = 1 on the other? 7. dououni-caicaira = 2 on the other? 8. coum-caicaira = 3 on the other? 9. dekai-caicaira = 4 on the other? 10. quinoi-da = 5 x 2. 11. cenai-ai-caibo = 1 + (the) hands. 12. dououni-ai-caibo = 2 + 10. 13. coum-ai-caibo = 3 + 10. 14. dekai-ai-caibo = 4 + 10. 15. quin-oibo = 5 x 3. 16. cenai-ai-quacoibo = 1 + 15. 17. dououni-ai-quacoibo = 2 + 15. 18. coum-ai-quacoibo = 3 + 15. 19. dekai-ai-quacoibo = 4 + 15. 20. quinoui-ai-quacoibo = 5 + 15.

The meanings assigned to the numerals 6 to 9 are entirely conjectural. They obviously mean 1, 2, 3, 4, taken a second time, and as the meanings I have given are often found in primitive systems, they have, at a venture, been given here.

LIFU, LOYALTY ISLANDS.[244]

1. ca. 2. lue. 3. koeni. 4. eke. 5. tji pi. 6. ca ngemen = 1 above. 7. lue ngemen = 2 above. 8. koeni ngemen = 3 above. 9. eke ngemen = 4 above. 10. lue pi = 2 x 5. 11. ca ko. 12. lue ko. 13. koeni ko. 14. eke ko. 15. koeni pi = 3 x 5. 16. ca huai ano. 17. lua huai ano. 18. koeni huai ano. 19. eke huai ano. 20. ca atj = 1 man.

BONGO.[245]

1. kotu. 2. ngorr. 3. motta. 4. neheo. 5. mui. 6. dokotu = [5] + 1. 7. dongorr = [5] + 2. 8. domotta = [5] + 3. 9. doheo = [5] + 4. 10. kih. 11. ki dokpo kotu = 10 + 1. 12. ki dokpo ngorr = 10 + 2. 13. ki dokpo motta = 10 + 3. 14. ki dokpo neheo = 10 + 4. 15. ki dokpo mui = 10 + 5. 16. ki dokpo mui do mui okpo kotu = 10 + 5 more, to 5, 1 more. 17. ki dokpo mui do mui okpo ngorr = 10 + 5 more, to 5, 2 more. 18. ki dokpo mui do mui okpo motta = 10 + 5 more, to 5, 3 more. 19. ki dokpo mui do mui okpo nehea = 10 + 5 more, to 5, 4 more. 20. mbaba kotu.

Above 20, the Lufu and the Bongo systems are vigesimal, so that they are, as a whole, mixed systems.

The Welsh scale begins as though it were to present a pure decimal structure, and no hint of the quinary element appears until it has passed 15. The Nahuatl, on the other hand, counts from 5 to 10 by the ordinary quinary method, and then appears to pass into the decimal form. But when 16 is reached, we find the quinary influence still persistent; and from this point to 20, the numeral words in both scales are such as to show that the notion of counting by fives is quite as prominent as the notion of referring to 10 as a base. Above 20 the systems become vigesimal, with a quinary or decimal structure appearing in all numerals except multiples of 20. Thus, in Welsh, 36 is unarbymtheg ar ugain, 1 + 5 + 10 + 20; and in Nahuatl the same number is cempualli caxtolli oce, 20 + 15 + 1. Hence these and similar number systems, though commonly alluded to as vigesimal, are really mixed scales, with 20 as their primary base. The Canaque scale differs from the Nahuatl only in forming a compound word for 15, instead of introducing a new and simple term.

In the examples which follow, it is not thought best to extend the lists of numerals beyond 10, except in special instances where the illustration of some particular point may demand it. The usual quinary scale will be found, with a few exceptions like those just instanced, to have the following structure or one similar to it in all essential details: 1, 2, 3, 4, 5, 5-1, 5-2, 5-3, 5-4, 10, 10-1, 10-2, 10-3, 10-4, 10-5, 10-5-1, 10-5-2, 10-5-3, 10-5-4, 20. From these forms the entire system can readily be constructed as soon as it is known whether its principal base is to be 10 or 20.

Turning first to the native African languages, I have selected the following quinary scales from the abundant material that has been collected by the various explorers of the "Dark Continent." In some cases the numerals of certain tribes, as given by one writer, are found to differ widely from the same numerals as reported by another. No attempt has been made at comparison of these varying forms of orthography, which are usually to be ascribed to difference of nationality on the part of the collectors.

FELOOPS.[246]

1. enory. 2. sickaba, or cookaba. 3. sisajee. 4. sibakeer. 5. footuck. 6. footuck-enory = 5-1. 7. footuck-cookaba = 5-2. 8. footuck-sisajee = 5-3. 9. footuck-sibakeer = 5-4. 10. sibankonyen.

KISSI.[247]

1. pili. 2. miu. 3. nga. 4. iol. 5. nguenu. 6. ngom-pum = 5-1. 7. ngom-miu = 5-2. 8. ngommag = 5-3. 9. nguenu-iol = 5-4. 10. to.

ASHANTEE.[248]

1. tah. 2. noo. 3. sah. 4. nah. 5. taw. 6. torata = 5 + 1. 7. toorifeenoo = 5 + 2. 8. toorifeessa = 5 + 3. 9. toorifeena = 5 + 4. 10. nopnoo.

BASA.[249]

1. do. 2. so. 3. ta. 4. hinye. 5. hum. 6. hum-le-do = 5 + 1. 7. hum-le-so = 5 + 2. 8. hum-le-ta = 5 + 3. 9. hum-le-hinyo = 5 + 4. 10. bla-bue.

JALLONKAS.[250]

1. kidding. 2. fidding. 3. sarra. 4. nani. 5. soolo. 6. seni. 7. soolo ma fidding = 5 + 2. 8. soolo ma sarra = 5 + 3. 9. soolo ma nani = 5 + 4. 10. nuff.

KRU.

1. da-do. 2. de-son. 3. de-tan. 4. de-nie. 5. de-mu. 6. dme-du = 5-1. 7. ne-son = [5] + 2. 8. ne-tan = [5] + 3. 9. sepadu = 10 - 1? 10. pua.

JALOFFS.[251]

1. wean. 2. yar. 3. yat. 4. yanet. 5. judom. 6. judom-wean = 5-1. 7. judom-yar = 5-2. 8. judom-yat = 5-3. 9. judom yanet = 5-4. 10. fook.

GOLO.[252]

1. mbali. 2. bisi. 3. bitta. 4. banda. 5. zonno. 6. tsimmi tongbali = 5 + 1. 7. tsimmi tobisi = 5 + 2. 8. tsimmi tobitta = 5 + 3. 9. tsimmi to banda = 5 + 4. 10. nifo.

FOULAH.[253]

1. go. 2. deeddee. 3. tettee. 4. nee. 5. jouee. 6. jego = 5-1. 7. jedeeddee = 5-2. 8. je-tettee = 5-3. 9. je-nee = 5-4. 10. sappo.

SOUSSOU.[254]

1. keren. 2. firing. 3. sarkan. 4. nani. 5. souli. 6. seni. 7. solo-fere = 5-2. 8. solo-mazarkan = 5 + 3. 9. solo-manani = 5 + 4. 10. fu.

BULLOM.[255]

1. bul. 2. tin. 3. ra. 4. hyul. 5. men. 6. men-bul = 5-1. 7. men-tin = 5-2. 8. men-ra = 5-3. 9. men-hyul = 5-4. 10. won.

VEI.[256]

1. dondo. 2. fera. 3. sagba. 4. nani. 5. soru. 6. sun-dondo = 5-1. 7. sum-fera = 5-2. 8. sun-sagba = 5-3. 9. sun-nani = 5-4. 10. tan.

DINKA.[257]

1. tok. 2. rou. 3. dyak. 4. nuan. 5. wdyets. 6. wdetem = 5-1. 7. wderou = 5-2. 8. bet, bed = 5-3. 9. wdenuan = 5-4. 10. wtyer = 5 x 2.

TEMNE.

1. in. 2. ran. 3. sas. 4. anle. 5. tr-amat. 6. tr-amat rok-in = 5 + 1. 7. tr-amat de ran = 5 + 2. 8. tr-amat re sas = 5 + 3. 9. tr-amat ro n-anle = 5 + 4. 10. tr-ofatr.

ABAKER.[258]

1. kili. 2. bore. 3. dotla. 4. ashe. 5. ini. 6. im kili = 5-1. 7. im-bone = 5-2. 8. ini-dotta = 5-3. 9. tin ashe = 5-4. 10. chica.

BAGRIMMA.[259]

1. kede. 2. sab. 3. muta. 4. so. 5. mi. 6. mi-ga = 5 + 1. 7. tsidi. 8. marta = 5 + 2. 9. do-so = [5] + 3 10. duk-keme.

PAPAA.[260]

1. depoo. 2. auwi. 3. ottong. 4. enne. 5. attong. 6. attugo. 7. atjuwe = [5] + 2. 8. attiatong = [5] + 3. 9. atjeenne = [5] + 4. 10. awo.

EFIK.[261]

1. kiet. 2. iba. 3. ita. 4. inan. 5. itiun. 6. itio-kiet = 5-1. 7. itia-ba = 5-2. 8. itia-eta = 5-3. 9. osu-kiet = 10 - 1? 10. duup.

NUPE.[262]

1. nini. 2. gu-ba. 3. gu-ta. 4. gu-ni. 5. gu-tsun. 6. gu-sua-yin = 5 + 1. 7. gu-tua-ba = 5 + 2. 8. gu-tu-ta = 5 + 3. 9. gu-tua-ni = 5 + 4. 10. gu-wo.

MOKKO.[263]

1. kiae. 2. iba. 3. itta. 4. inan. 5. uettin. 6. itjueekee = 5 + 1. 7. ittiaba = 5 + 2. 8. itteiata = 5 + 3. 9. huschukiet. 10. bueb.

KANURI.[264]

1. tilo. 2. ndi. 3. yasge. 4. dege. 5. ugu. 6. arasge = 5 + 1. 7. tulur. 8. wusge = 5 + 3. 9. legar. 10. megu = 2 x 5.

BININ.[265]

1. bo. 2. be. 3. la. 4. nin. 5. tang. 6. tahu = 5 + 1? 7. tabi = 5 + 2. 8. tara = 5 + 3. 9. ianin (tanin?) = 5 + 4? 10. te.

KREDY.[266]

1. baia. 2. rommu. 3. totto. 4. sosso. 5. saya. 6. yembobaia = [5] + 1. 7. yemborommu = [5] + 2. 8. yembototto = [5] + 3. 9. yembososso = [5] + 4. 10. puh.

HERERO.[267]

1. mue. 2. vari. 3. tatu. 4. ne. 5. tano. 6. hambou-mue = [5] + 1. 7. hambou-vari = [5] + 2. 8. hambou-tatu = [5] + 3. 9. hambou-ne = [5] + 4. 10.

KI-YAU.[268]

1. jumo. 2. wawiri. 3. watatu. 4. mcheche. 5. msano. 6. musano na jumo = 5 + 1. 7. musano na wiri = 5 + 2. 8. musano na watatu = 5 + 3. 9. musano na mcheche = 5 + 4. 10. ikumi.

FERNANDO PO.[269]

1. muli. 2. mempa. 3. meta. 4. miene. 5. mimito. 6. mimito na muli = 5 + 1. 7. mimito na mempa = 5 + 2. 8. mimito na meta = 5 + 3. 9. mimito na miene = 5 + 4. 10. miemieu = 5-5?

KI-NYASSA

1. kimodzi. 2. vi-wiri. 3. vi-tatu. 4. vinye. 5. visano. 6. visano na kimodzi = 5 + 1. 7. visano na vi-wiri = 5 + 2. 8. visano na vitatu = 5 + 3. 9. visano na vinye = 5 + 4. 10. chikumi.

BALENGUE.[270]

1. guevoho. 2. ibare. 3. raro. 4. inai. 5. itano. 6. itano na guevoho = 5 + 1. 7. itano na ibare = 5 + 2. 8. itano na raro = 5 + 3. 9. itano na inai = 5 + 4. 10. ndioum, or nai-hinai.

KUNAMA.[271]

1. ella. 2. bare. 3. sadde. 4. salle. 5. kussume. 6. kon-t'-ella = hand 1. 7. kon-te-bare = hand 2. 8. kon-te-sadde = hand 3. 9. kon-te-salle = hand 4. 10. kol-lakada.

GOLA.[272]

1. ngoumou. 2. ntie. 3. ntai. 4. tina. 5. nonon. 6. diegoum = [5] + 1. 7. dientie = [5] + 2. 8. dietai = [5] + 3. 9. dectina = [5] + 4. 10. esia.

BAREA.[273]

1. doko 2. arega. 3. sane. 4. sone. 5. oita. 6. data. 7. dz-ariga = 5 + 2. 8. dis-sena = 5 + 3. 9. lefete-mada = without 10. 10. lefek.

MATIBANI.[274]

1. mosa. 2. pili. 3. taru. 4. teje. 5. taru. 6. tana mosa = 5-1. 7. tana pili = 5-2. 8. tana taru = 5-3. 9. loco. 10. loco nakege.

BONZE.[275]

1. tan. 2. vele. 3. daba. 4. nani. 5. lolou. 6. maida = [5] + 1. 7. maifile = [5] + 2. 8. maishaba = [5] + 3. 9. mainan = [5] + 4. 10. bou.

MPOVI

1. moueta. 2. bevali. 3. betata. 4. benai. 5. betani. 6. betani moueta = 5-1. 7. betani bevali = 5-2. 8. betani betata = 5-3. 9. betani benai = 5-4. 10. nchinia.

TRITON'S BAY, NEW QUINEA.[276]

1. samosi. 2. roueti. 3. tourou. 4. faat. 5. rimi. 6. rim-samosi = 5-1. 7. rim-roueti = 5-2. 8. rim-tourou = 5-3. 9. rim-faat = 5-4. 10. outsia.

ENDE, OR FLORES.[277]

1. sa. 2. zua. 3. telu. 4. wutu. 5. lima = hand. 6. lima-sa = 5-1, or hand 1. 7. lima-zua = 5-2. 8. rua-butu = 2 x 4? 9. trasa = [10] - 1? 10. sabulu.

MALLICOLO.[278]

1. tseekaee. 2. ery. 3. erei. 4. ebats. 5. ereem. 6. tsookaee = [5] + 1. 7. gooy = [5] + 2. 8. hoorey = [5] + 3. 9. goodbats = [5] + 4. 10. senearn.

EBON, MARSHALL ISLANDS.[279]

1. iuwun. 2. drud. 3. chilu. 4. emer. 5. lailem. 6. chilchinu = 5 + 1. 7. chilchime = 5 + 2. 8. twalithuk = [10] - 2. 9. twahmejuwou = [10] - 1. 10. iungou.

UEA, LOYALTY ISLAND.[280]

1. tahi. 2. lua. 3. tolu. 4. fa. 5. lima. 6. tahi. 7. lua. 8. tolu. 9. fa. 10. lima.

UEA.[280]—[another dialect.]

1. hacha. 2. lo. 3. kuun. 4. thack. 5. thabumb. 6. lo-acha = 2d 1. 7. lo-alo = 2d 2. 8. lo-kuun = 2d 3. 9. lo-thack = 2d 4. 10. lebenetee.

ISLE OF PINES.[281]

1. ta. 2. bo. 3. beti. 4. beu. 5. ta-hue. 6. no-ta = 2d 1. 7. no-bo = 2d 2. 8. no-beti = 2d 3. 9. no-beu = 2d 4. 10. de-kau.

UREPARAPARA, BANKS ISLANDS.[282]

1. vo towa. 2. vo ro. 3. vo tol. 4. vo vet. 5. teveliem = 1 hand. 6. leve jea = other 1. 7. leve ro = other 2. 8. leve tol = other 3. 9. leve vet = other 4. 10. sanowul = 2 sets.

MOTA, BANKS ISLANDS.[282]

1. tuwale. 2. nirua. 3. nitol. 4. nivat. 5. tavelima = 1 hand. 6. laveatea = other 1. 7. lavearua = other 2. 8. laveatol = other 3. 9. laveavat = other 4. 10. sanavul = 2 sets.

NEW CALEDONIA.[283]

1. parai. 2. paroo. 3. parghen. 4. parbai. 5. panim. 6. panim-gha = 5-1. 7. panim-roo = 5-2. 8. panim-ghen = 5-3. 9. panim-bai = 5-4. 10. parooneek.

YENGEN, NEW CAL.[284]

1. hets. 2. heluk. 3. heyen. 4. pobits. 5. nim = hand. 6. nim-wet = 5-1. 7. nim-weluk = 5-2. 8. nim-weyen = 5-3. 9. nim-pobit = 5-4. 10. pain-duk.

ANEITEUM.[285]

1. ethi. 2. ero. 3. eseik. 4. manohwan. 5. nikman. 6. nikman cled et ethi = 5 + 1. 7. nikman cled et oro = 5 + 2. 8. nikman cled et eseik = 5 + 3. 9. nikman cled et manohwan = 5 + 4. 10. nikman lep ikman = 5 + 5.

TANNA

1. riti. 2. karu. 3. kahar. 4. kefa. 5. krirum. 6. krirum riti = 5-1. 7. krirum karu = 5-2. 8. krirum kahar? = 5-3. 9. krirum kefa? = 5-4. 10. ——

EROMANGA

1. sai. 2. duru. 3. disil. 4. divat. 5. siklim = 1 hand. 6. misikai = other 1? 7. siklim naru = 5-2. 8. siklim disil = 5-3. 9. siklim mindivat = 5 + 4. 10. narolim = 2 hands.

FATE, NEW HEB.[286]

1. iskei. 2. rua. 3. tolu. 4. bate. 5. lima = hand. 6. la tesa = other 1. 7. la rua = other 2. 8. la tolu = other 3. 9. la fiti = other 4. 10. relima = 2 hands.

API, NEW HEB.

1. tai. 2. lua. 3. tolu. 4. vari. 5. lima = hand. 6. o rai = other 1. 7. o lua = other 2. 8. o tolo = other 3. 9. o vari = other 4. 10. lua lima = 2 hands.

SESAKE, NEW HEB.

1. sikai. 2. dua. 3. dolu. 4. pati. 5. lima = hand. 6. la tesa = other 1. 7. la dua = other 2. 8. la dolu = other 3. 9. lo veti = other 4. 10. dua lima = 2 hands.

PAMA, NEW HEB.

1. tai. 2. e lua. 3. e tolu. 4. e hati. 5. e lime = hand. 6. a hitai = other 1. 7. o lu = other 2. 8. o tolu = other 3. 9. o hati = other 4. 10. ha lua lim = 2 hands

AURORA, NEW HEB.

1. tewa. 2. i rua. 3. i tol. 4. i vat. 5. tavalima = 1 hand. 6. lava tea = other 1. 7. lava rua = other 2. 8. lava tol = other 3. 9. la vat = other 4. 10. sanwulu = two sets.

TOBI.[287]

1. yat. 2. glu. 3. ya. 4. uan. 5. yanim = 1 hand. 6. yawor = other 1. 7. yavic = other 2. 8. yawa = other 3. 9. yatu = other 4. 10. yasec.

PALM ISLAND.[288]

1. yonkol. 2. yakka. 3. tetjora. 4. tarko. 5. yonkol mala = 1 hand.

JAJOWERONG, VICTORIA.[288]

1. kiarp. 2. bulaits. 3. bulaits kiarp = 2-1. 4. bulaits bulaits = 2-2. 5. kiarp munnar = 1 hand. 6. bulaits bulaits bulaits = 2-2-2. 10. bulaits munnar = 2 hands.

The last two scales deserve special notice. They are Australian scales, and the former is strongly binary, as are so many others of that continent. But both show an incipient quinary tendency in their names for 5 and 10.

CAMBODIA.[289]

1. muy. 2. pir. 3. bey. 4. buon. 5. pram. 6. pram muy = 5-1. 7. pram pil = 5-2. 8. pram bey = 5-3. 9. pram buon = 5-4. 10. dap.

TSCHUKSCHI.[290]

1. inen. 2. nirach. 3. n'roch. 4. n'rach. 5. miligen = hand. 6. inen miligen = 1-5. 7. nirach miligen = 2-5. 8. anwrotkin. 9. chona tsinki. 10. migitken = both hands.

KOTTISCH[291]

1. hutsa. 2. ina. 3. tona. 4. sega. 5. chega. 6. chelutsa = 5 + 1. 7. chelina = 5 + 2. 8. chaltona = 5 + 3. 9. tsumnaga = 10 - 1. 10. haga.

1. a towshek. 2. hipah, or malho. 3. pingishute. 4. sesaimat. 5. talema. 6. okvinile, or ahchegaret = another 1? 7. talema-malronik = 5-two of them. 8. pingishu-okvingile = 2d 3? 9. kolingotalia = 10 - 1? 10. koleet.

KAMTSCHATKA, SOUTH.[293]

1. dischak. 2. kascha. 3. tschook. 4. tschaaka. 5. kumnaka. 6. ky'lkoka. 7. itatyk = 2 + 5. 8. tschookotuk = 3 + 5. 9. tschuaktuk = 4 + 5. 10. kumechtuk = 5 + 5.

ALEUTS[294]

1. ataqan. 2. aljak. 3. qankun. 4. sitsin. 5. tsan = my hand. 6. atun = 1 + 5. 7. ulun = 2 + 5. 8. qamtsin = 3 + 5. 9. sitsin = 4 + 5. 10. hatsiq.

TCHIGLIT, MACKENZIE R.[295]

1. ataotcirkr. 2. aypak, or malloerok. 3. illaak, or pinatcut. 4. tcitamat. 5. tallemat. 6. arveneloerit. 7. arveneloerit-aypak = 5 + 2. 8. arveneloerit-illaak = 5 + 3. 9. arveneloerit-tcitamat = 5 + 4. 10. krolit.

SAHAPTIN (NEZ PERCES).[296]

1. naks. 2. lapit. 3. mitat. 4. pi-lapt = 2 x 2. 5. pachat. 6. oi-laks = [5] + 1. 7. oi-napt = [5] + 2. 8. oi-matat = [5] + 3. 9. koits. 10. putimpt.

GREENLAND.[297]

1. atauseq. 2. machdluq. 3. pinasut. 4. sisamat 5. tadlimat. 6. achfineq-atauseq = other hand 1. 7. achfineq-machdluq = other hand 2. 8. achfineq-pinasut = other hand 3. 9. achfineq-sisamat = other hand 4. 10. qulit. 11. achqaneq-atauseq = first foot 1. 12. achqaneq-machdluq = first foot 2. 13. achqaneq-pinasut = first foot 3. 14. achqaneq-sisamat = first foot 4. 15. achfechsaneq? 16. achfechsaneq-atauseq = other foot 1. 17. achfechsaneq-machdlup = other foot 2. 18. achfechsaneq-pinasut = other foot 3. 19. achfechsaneq-sisamat = other foot 4. 20. inuk navdlucho = a man ended.

Up to this point the Greenlander's scale is almost purely quinary. Like those of which mention was made at the beginning of this chapter, it persists in progressing by fives until it reaches 20, when it announces a new base, which shows that the system will from now on be vigesimal. This scale is one of the most interesting of which we have any record, and will be noticed again in the next chapter. In many respects it is like the scale of the Point Barrow Eskimo, which was given early in Chapter III. The Eskimo languages are characteristically quinary-vigesimal in their number systems, but few of them present such perfect examples of that method of counting as do the two just mentioned.

CHIPPEWAY.[298]

1. bejig. 2. nij. 3. nisswi. 4. niwin. 5. nanun. 6. ningotwasswi = 1 again? 7. nijwasswi = 2 again? 8. nishwasswi = 3 again? 9. jangasswi = 4 again? 10. midasswi = 5 again.

MASSACHUSETTS.[299]

1. nequt. 2. neese. 3. nish. 4. yaw. 5. napanna = on one side, i.e. 1 hand. 6. nequttatash = 1 added. 7. nesausuk = 2 again? 8. shawosuk = 3 again? 9. pashoogun = it comes near, i.e. to 10. 10. puik.

OJIBWA OF CHEGOIMEGON.[300]

1. bashik. 2. neensh. 3. niswe. 4. newin. 5. nanun. 6. ningodwaswe = 1 again? 7. nishwaswe = 2 again? 8. shouswe = 3 again? 9. shangaswe = 4 again? 10. medaswe = 5 again?

OTTAWA.

1. ningotchau. 2. ninjwa. 3. niswa. 4. niwin. 5. nanau. 6. ningotwaswi = 1 again? 7. ninjwaswi = 2 again? 8. nichwaswi = 3 again? 9. shang. 10. kwetch.

DELAWARE.

1. n'gutti. 2. niskha. 3. nakha. 4. newa. 5. nalan [akin to palenach, hand]. 6. guttash = 1 on the other side. 7. nishash = 2 on the other side. 8. khaash = 3 on the other side. 9. peshgonk = coming near. 10. tellen = no more.

SHAWNOE.

1. negote. 2. neshwa. 3. nithuie. 4. newe. 5. nialinwe = gone. 6. negotewathwe = 1 further. 7. neshwathwe = 2 further. 8. sashekswa = 3 further? 9. chakatswe [akin to chagisse, "used up"]. 10. metathwe = no further.

MICMAC.[301]

1. naiookt. 2. tahboo. 3. seest. 4. naioo. 5. nahn. 6. usoo-cum. 7. eloo-igunuk. 8. oo-gumoolchin. 9. pescoonaduk. 10. mtlin.

One peculiarity of the Micmac numerals is most noteworthy. The numerals are real verbs, instead of adjectives, or, as is sometimes the case, nouns. They are conjugated through all the variations of mood, tense, person, and number. The forms given above are not those that would be used in counting, but are for specific use, being varied according to the thought it was intended to express. For example, naiooktaich = there is 1, is present tense; naiooktaichcus, there was 1, is imperfect; and encoodaichdedou, there will be 1, is future. The variation in person is shown by the following inflection:

PRESENT TENSE.

1st pers. tahboosee-ek = there are 2 of us. 2d pers. tahboosee-yok = there are 2 of you. 3d pers. tahboo-sijik = there are 2 of them.

IMPERFECT TENSE.

1st pers. tahboosee-egup = there were 2 of us. 2d pers. tahboosee-yogup = there were 2 of you. 3d pers. tahboosee-sibunik = there were 2 of them.

FUTURE TENSE.

3d pers. tahboosee-dak = there will be 2 of them, etc.

The negative form is also comprehended in the list of possible variations. Thus, tahboo-seekw, there are not 2 of them; mah tahboo-seekw, there will not be 2 of them; and so on, through all the changes which the conjugation of the verb permits.

OLD ALGONQUIN.

1. peygik. 2. ninsh. 3. nisswey. 4. neyoo. 5. nahran = gone. 6. ningootwassoo = 1 on the other side. 7. ninshwassoo = 2 on the other side. 8. nisswasso = 3 on the other side. 9. shangassoo [akin to chagisse, "used up"]. 10. mitassoo = no further.

OMAHA.

1. meeachchee. 2. nomba. 3. rabeenee. 4. tooba. 5. satta = hand, i.e. all the fingers turned down. 6. shappai = 1 more. 7. painumba = fingers 2. 8. pairabeenee = fingers 3. 9. shonka = only 1 finger (remains). 10. kraibaira = unbent.[302]

CHOCTAW.

1. achofee. 2. tuklo. 3. tuchina. 4. ushta. 5. tahlape = the first hand ends. 6. hanali. 7. untuklo = again 2. 8. untuchina = again 3. 9. chokali = soon the end; i.e. next the last. 10. pokoli.

1. kouanigh. 2. behit. 3. daho. 4. hehweh. 5. dihsehkon. 6. dunkeh. 7. bisekah = 5 + 2. 8. dousehka = 5 + 3. 9. hehwehsehka = 4 + hand. 10. behnehaugh.

CHIPPEWAY.

1. payshik. 2. neesh. 3. neeswoy. 4. neon. 5. naman = gone. 6. nequtwosswoy = 1 on the other side. 7. neeshswosswoy = 2 on the other side. 8. swoswoy = 3 on the other side? 9. shangosswoy [akin to chagissi, "used up"]. 10. metosswoy = no further.

1. nancas. 2. nass. 3. colle. 4. tacache. 5. seppacan. 6. pacanancus = 5 + 1. 7. pacaness = 5 + 2. 8. pacalcon = 5 + 3. 9. sickinish = hands minus? 10. neusne.

PAWNEE.

1. askoo. 2. peetkoo. 3. touweet. 4. shkeetiksh. 5. sheeooksh = hands half. 6. sheekshabish = 5 + 1. 7. peetkoosheeshabish = 2 + 5. 8. touweetshabish = 3 + 5. 9. looksheereewa = 10 - 1. 10. looksheeree = 2d 5?

MINSI.

1. gutti. 2. niskha. 3. nakba. 4. newa. 5. nulan = gone? 6. guttash = 1 added. 7. nishoash = 2 added. 8. khaash = 3 added. 9. noweli. 10. wimbat.

KONLISCHEN.

1. tlek. 2. tech. 3. nezk. 4. taakun. 5. kejetschin. 6. klet uschu = 5 + 1. 7. tachate uschu = 5 + 2. 8. nesket uschu = 5 + 3. 9. kuschok = 10 - 1? 10. tschinkat.

TLINGIT.[303]

1. tlek. 2. deq. 3. natsk. 4. dak'on = 2d 2. 5. kedjin = hand. 6. tle durcu = other 1. 7. daqa durcu = other 2. 8. natska durcu = other 3. 9. gocuk. 10. djinkat = both hands.

RAPID, OR FALL, INDIANS.

1. karci. 2. neece. 3. narce. 4. nean. 5. yautune. 6. neteartuce = 1 over? 7. nesartuce = 2 over? 8. narswartuce = 3 over? 9. anharbetwartuce = 4 over? 10. mettartuce = no further?

HEILTSUK.[304]

1. men. 2. matl. 3. yutq. 4. mu. 5. sky'a. 6. katla. 7. matlaaus = other 2? 8. yutquaus = other 3? 9. mamene = 10 - 1. 10. aiky'as.

NOOTKA.[305]

1. nup. 2. atla. 3. katstsa. 4. mo. 5. sutca. 6. nopo = other 1? 7. atlpo = other 2? 8. atlakutl = 10 - 2. 9. ts'owakutl = 10 - 1. 10. haiu.

TSIMSHIAN.[306]

1. gyak. 2. tepqat. 3. guant. 4. tqalpq. 5. kctonc (from anon, hand). 6. kalt = 2d 1. 7. t'epqalt = 2d 2. 8. guandalt = 2d 3? 9. kctemac. 10. gy'ap.

BILQULA.[306]

1. (s)maotl. 2. tlnos. 3. asmost. 4. mos. 5. tsech. 6. tqotl = 2d 1? 7. nustlnos = 2d 2? 8. k'etlnos = 2 x 4. 9. k'esman. 10. tskchlakcht.

MOLELE.[307]

1. mangu. 2. lapku. 3. mutka. 4. pipa. 5. pika. 6. napitka = 1 + 5. 7. lapitka = 2 + 5. 8. mutpitka = 3 + 5. 9. laginstshiatkus. 10. nawitspu.

WAIILATPU.[308]

1. na. 2. leplin. 3. matnin. 4. piping. 5. tawit. 6. noina = [5] + 1. 7. noilip = [5] + 2. 8. noimat = [5] + 3. 9. tanauiaishimshim. 10. ningitelp.

LUTUAMI.[307]

1. natshik. 2. lapit. 3. ntani. 4. wonip. 5. tonapni. 6. nakskishuptane = 1 + 5. 7. tapkishuptane = 2 + 5. 8. ndanekishuptane = 3 + 5. 9. natskaiakish = 10 - 1. 10. taunip.

SASTE (SHASTA).[309]

1. tshiamu. 2. hoka. 3. hatski. 4. irahaia. 5. etsha. 6. tahaia. 7. hokaikinis = 2 + 5. 8. hatsikikiri = 3 + 5. 9. kirihariki-ikiriu. 10. etsehewi.

CAHUILLO.[310]

1. supli. 2. mewi. 3. mepai. 4. mewittsu. 5. nomekadnun. 6. kadnun-supli = 5-1. 7. kan-munwi = 5-2. 8. kan-munpa = 5-3. 9. kan-munwitsu = 5-4. 10. nomatsumi.

TIMUKUA.[311]

1. yaha. 2. yutsa. 3. hapu. 4. tseketa. 5. marua. 6. mareka = 5 + 1 7. pikitsa = 5 + 2 8. pikinahu = 5 + 3 9. peke-tsaketa = 5 + 4 10. tuma.

OTOMI[312]

1. nara. 2. yocho. 3. chiu. 4. gocho. 5. kuto. 6. rato = 1 + 5. 7. yoto = 2 + 5. 8. chiato = 3 + 5. 9. guto = 4 + 5. 10. reta.

TARASCO.[313]

1. ma. 2. dziman. 3. tanimo. 4. tamu. 5. yumu. 6. kuimu. 7. yun-dziman = [5] + 2. 8. yun-tanimo = [5] + 3. 9. yun-tamu = [5] + 4. 10. temben.

MATLALTZINCAN.[314]

1. indawi. 2. inawi. 3. inyuhu. 4. inkunowi. 5. inkutaa. 6. inda-towi = 1 + 5. 7. ine-towi = 2 + 5. 8. ine-ukunowi = 2-4. 9. imuratadahata = 10 - 1? 10. inda-hata.

CORA.[315]

1. ceaut. 2. huapoa. 3. huaeica. 4. moacua. 5. anxuvi. 6. a-cevi = [5] + 1. 7. a-huapoa = [5] + 2. 8. a-huaeica = [5] + 3. 9. a-moacua = [5] + 4. 10. tamoamata (akin to moamati, "hand").

AYMARA.[316]

1. maya. 2. paya. 3. kimsa. 4. pusi. 5. piska. 6. tsokta. 7. pa-kalko = 2 + 5. 8. kimsa-kalko = 3 + 5. 9. pusi-kalko = 4 + 5. 10. tunka.

CARIBS OF ESSEQUIBO, GUIANA.[317]

1. oween. 2. oko. 3. oroowa. 4. oko-baimema. 5. wineetanee = 1 hand. 6. owee-puimapo = 1 again? 7. oko-puimapo = 2 again? 8. oroowa-puimapo = 3 again? 9. oko-baimema-puimapo = 4 again? 10. oween-abatoro.

CARIB.[318] (ROUCOUYENNE?)

1. aban, amoin. 2. biama. 3. eleoua. 4. biam-bouri = 2 again? 5. ouacabo-apourcou-aban-tibateli. 6. aban laoyagone-ouacabo-apourcou. 7. biama laoyagone-ouacabo-apourcou. 8. eleoua laoyagone-ouacabo-apourcou. 9. —— 10. chon noucabo.

It is unfortunate that the meanings of these remarkable numerals cannot be given. The counting is evidently quinary, but the terms used must have been purely descriptive expressions, having their origin undoubtedly in certain gestures or finger motions. The numerals obtained from this region, and from the tribes to the south and east of the Carib country, are especially rich in digital terms, and an analysis of the above numerals would probably show clearly the mental steps through which this people passed in constructing the rude scale which served for the expression of their ideas of number.

KIRIRI.[319]

1. biche. 2. watsani. 3. watsani dikie. 4. sumara oroba. 5. mi biche misa = 1 hand. 6. mirepri bu-biche misa sai. 7. mirepri watsani misa sai. 8. mirepri watsandikie misa sai. 9. mirepri sumara oraba sai. 10. mikriba misa sai = both hands.

CAYUBABA[320]

1. pebi. 2. mbeta. 3. kimisa. 4. pusi. 5. pisika. 6. sukuta. 7. pa-kaluku = 2 again? 8. kimisa-kaluku = 3 again? 9. pusu-kaluku = 4 again? 10. tunka.

SAPIBOCONA[320]

1. karata. 2. mitia. 3. kurapa. 4. tsada. 5. maidara (from arue, hand). 6. karata-rirobo = 1 hand with. 7. mitia-rirobo = 2 hand with. 8. kurapa-rirobo = 3 hand with. 9. tsada-rirobo = 4 hand with. 10. bururutse = hand hand.

TICUNA.[321]

1. hueih. 2. tarepueh. 3. tomepueh. 4. aguemoujih 5. hueamepueh. 6. naimehueapueh = 5 + 1. 7. naimehueatareh = 5 + 2. 8. naimehueatameapueh = 5 + 3. 9. gomeapueh = 10 - 1. 10. gomeh.

YANUA.[322]

1. tckini. 2. nanojui. 3. munua. 4. nairojuino = 2d 2. 5. tenaja. 6. teki-natea = 1 again? 7. nanojui-natea = 2 again? 8. munua-natea = 3 again? 9. nairojuino-natea = 4 again? 10. huijejuino = 2 x 5?

The foregoing examples will show with considerable fulness the wide dispersion of the quinary scale. Every part of the world contributes its share except Europe, where the only exceptions to the universal use of the decimal system are the half-dozen languages, which still linger on its confines, whose number base is the vigesimal. Not only is there no living European tongue possessing a quinary number system, but no trace of this method of counting is found in any of the numerals of the earlier forms of speech, which have now become obsolete. The only possible exceptions of which I can think are the Greek [Greek: pempazein], to count by fives, and a few kindred words which certainly do hint at a remote antiquity in which the ancestors of the Greeks counted on their fingers, and so grouped their units into fives. The Roman notation, the familiar I., II., III., IV. (originally IIII.), V., VI., etc., with equal certainty suggests quinary counting, but the Latin language contains no vestige of anything of the kind, and the whole range of Latin literature is silent on this point, though it contains numerous references to finger counting. It is quite within the bounds of possibility that the prehistoric nations of Europe possessed and used a quinary numeration. But of these races the modern world knows nothing save the few scanty facts that can be gathered from the stone implements which have now and then been brought to light. Their languages have perished as utterly as have the races themselves, and speculation concerning them is useless. Whatever their form of numeration may have been, it has left no perceptible trace on the languages by which they were succeeded. Even the languages of northern and central Europe which were contemporary with the Greek and Latin of classical times have, with the exception of the Celtic tongues of the extreme North-west, left behind them but meagre traces for the modern student to work on. We presume that the ancient Gauls and Goths, Huns and Scythians, and other barbarian tribes had the same method of numeration that their descendants now have; and it is a matter of certainty that the decimal scale was, at that time, not used with the universality which now obtains; but wherever the decimal was not used, the universal method was vigesimal; and that the quinary ever had anything of a foothold in Europe is only to be guessed from its presence to-day in almost all of the other corners of the world.

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.

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.

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:

ABKHASIA.[349]

10. zpha-ba. 20. gphozpha = 2 x 10. 30. gphozphei zphaba = 20 + 10. 40. gphin-gphozpha = 2 x 20. 60. chin-gphozpha = 3 x 20. 80. phsin-gphozpha = 4 x 20. 100. sphki.

AVARI

10. antsh-go. 20. qo-go. 30. lebergo. 40. khi-qogo = 2 x 20. 50. khiqojalda antshgo = 40 + 10. 60. lab-qogo = 3 x 20. 70. labqojalda antshgo = 60 + 10. 80. un-qogo = 4 x 20. 100. nusgo.

KURI

10. tshud. 20. chad. 30. channi tshud = 20 + 10. 40. jachtshur. 50. jachtshurni tshud = 40 + 10. 60. put chad = 3 x 20. 70. putchanni tshud = 60 + 10. 80. kud-chad = 4 x 20. 90. kudchanni tshud = 80 + 10. 100. wis.

UDI

10. witsh. 20. qa. 30. sa-qo-witsh = 20 + 10. 40. pha-qo = 2 x 20. 50. pha-qo-witsh = 40 + 10. 60. chib-qo = 3 x 20. 70. chib-qo-witsh = 60 + 10. 80. bip-qo = 4 x 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 x 20. 50. sauz-tqe ith = 40 + 10. 60. chuz-tqa = 3 x 20. 70. chuz-tqe ith = 60 + 10. 80. w-iez-tqa = 4 x 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 x 20. 50. sauz-tqa-itt = 40 + 10. 60. chouz-tq = 3 x 20. 70. chouz-tqa-itt = 60 + 10. 80. dhewuz-tq = 4 x 20. 90. dhewuz-tqa-itt = 80 + 10. 100. phchauz-tq = 5 x 20. 200. itsha-tq = 10 x 20. 300. phehiitsha-tq = 15 x 20. 1000. satsh tqauz-tqa itshatqa = 2 x 20 x 20 + 200.

GEORGIA

10. athi. 20. otsi. 30. ots da athi = 20 + 10. 40. or-m-otsi = 2 x 20. 50. ormots da athi = 40 + 10. 60. sam-otsi = 3 x 20. 70. samots da athi = 60 + 10. 80. othch-m-otsi = 4 x 20. 90. othmots da athi = 80 + 10. 100. asi. 1000. ath-asi = 10 x 100.

LAZI

10. wit. 20. oets. 30. oets do wit = 20 x 10. 40. dzur en oets = 2 x 20. 50. dzur en oets do wit = 40 + 10. 60. dzum en oets = 3 x 20. 70. dzum en oets do wit = 60 + 10. 80. otch-an-oets = 4 x 20. 100. os. 1000. silia (akin to Greek).

CHUNSAG.[350]

10. ants-go. 20. chogo. 30. chogela antsgo = 20 + 10. 40. kichogo = 2 x 20. 50. kichelda antsgo = 40 + 10. 60. taw chago = 3 x 20. 70. taw chogelda antsgo = 60 + 10. 80. uch' chogo = 4 x 20. 90. uch' chogelda antsgo. 100. nusgo. 1000. asargo (akin to Persian).

DIDO.[351]

10. zino. 20. ku. 30. kunozino. 40. kaeno ku = 2 x 20. 50. kaeno kuno zino = 40 + 10. 60. sonno ku = 3 x 20. 70. sonno kuno zino = 60 + 10. 80. uino ku = 4 x 20. 90. uino huno zino = 80 + 10. 100. bischon. 400. kaeno kuno zino = 40 x 10.

AKARI

10. entzelgu. 20. kobbeggu. 30. lowergu. 40. kokawu = 2 x 20. 50. kikaldanske = 40 + 10. 60. secikagu. 70. kawalkaldansku = 3 x 20 + 10. 80. onkuku = 4 x 20. 90. onkordansku = 4 x 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 x 10. 50. ptl'isch-era-pschirre = 40 + 10. 60. chi-tsch = 6 x 10. 70. chitsch-era-pschirre = 60 + 10. 80. toshitl = 20 x 4? 90. toshitl-era-pschirre = 80 + 10. 100. scheh. 1000. min (Tartar) or schi-psche = 100 x 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 x 20. 100. milin chlikkin = 5 x 20. 200. mingit chlikkin = 10 x 20, i.e. 10 men. 1000. miligen chlin-chlikkin = 5 x 200, i.e. five (times) 10 men.

AINO.[353]

10. wambi. 20. choz. 30. wambi i-doehoz = 10 from 40. 40. tochoz = 2 x 20. 50. wambi i-richoz = 10 from 60. 60. rechoz = 3 x 20. 70. wambi i inichoz = 10 from 80. 80. inichoz = 4 x 20. 90. wambi aschikinichoz = 10 from 100. 100. aschikinichoz = 5 x 20. 110. wambi juwanochoz = 10 from 120. 120. juwano choz = 6 x 20. 130. wambi aruwanochoz = 10 from 140. 140. aruwano choz = 7 x 20. 150. wambi tubischano choz = 10 from 160. 160. tubischano choz = 8 x 20. 170. wambi schnebischano choz = 10 from 180. 180. schnebischano choz = 9 x 20. 190. wambi schnewano choz = 10 from 200. 200. schnewano choz = 10 x 20. 300. aschikinichoz i gaschima chnewano choz = 5 x 20 + 10 x 20. 400. toschnewano choz = 2 x (10 x 20). 500. aschikinichoz i gaschima toschnewano choz = 100 + 400. 600. reschiniwano choz = 3 x 200. 700. aschikinichoz i gaschima reschiniwano choz = 100 + 600. 800. inischiniwano choz = 4 x 200. 900. aschikinichoz i gaschima inischiniwano choz = 100 + 800. 1000. aschikini schinewano choz = 5 x 200. 2000. wanu schinewano choz = 10 x (10 x 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 x 10, or kha nat = 2 x 20. 50. kafano = 5 x 10, or kha nat sa kati = 2 x 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 x 20. 60. api hisi = 3 x 20. 80. upun hisi = 4 x 20. 100. mone hisi = 5 x 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 x 20. 40. an-inai = 2 x 20. 100. tain-inai = 5 x 20. 100. tanai-inai = 5 x 20. 200. teya-inai = 10 x 20. 200. sham-inai = 10 x 20. 300. teya-tain-inai 300. heang-tanai-inai = (10 + 5) x 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:

10. koleet. 20. enuenok. 30. enuenok kolinik = 20 + 10. 40. malho kepe ak = 2 x 20. 50. malho-kepe ak-kolmik che pah ak to = 2 x 20 + 10. 60. pingi shu-kepe ak = 3 x 20. 100. tale ma-kepe ak = 5 x 20. 400. enue nok ke pe ak = 20 x 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 pinatcunik 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 x 10. 40. dak'on djinkat = 4 x 10. 50. kedjin djinkat = 5 x 10. 60. tle durcu djinkat = 6 x 10. 70. daqa durcu djinkat = 7 x 10. 80. natska durcu djinkat = 8 x 10. 90. gocuk durcu djinkat = 9 x 10. 100. kedjin ka = 5 men, or 5 x 20. 200. djinkat ka = 10 x 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 x 20. 60. katstsek = 3 x 20. 80. moyek = 4 x 20. 100. sutc'ek = 5 x 20. 120. nop'ok = 6 x 20. 140. atlpok = 7 x 20. 160. atlakutlek = 8 x 20. 180. ts'owakutlek = 9 x 20. 200. haiuk = 10 x 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 x 5. 40. taiguaco ajte = 20 x 2. 50. taiguaco ajte co juyopamauj ajte = 20 x 2 + 5 x 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 x 30. 50. yote-m'a-ratta = 2 x 20 + 10. 60. hiu-te = 3 x 20. 70. hiute-m'a-ratta = 3 x 20 + 10. 80. gooho-rate = 4 x 20. 90. gooho-rate-m'a ratta = 4 x 20 + 10. 100. cytta-te = 5 x 20, or nanthebe = 1 x 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 x 20. 50. lahuyoxkal = 60 - 10. 60. oxkal = 3 x 20. 70. lahucankal = 80 - 10. 80. cankal = 4 x 20. 90. lahuyokal = 100 - 10. 100. hokal = 5 x 20. 110. lahu uackal = 120 - 10. 120. uackal = 6 x 20. 130. lahu uuckal = 140 - 10. 140. uuckal = 7 x 20. 200. lahuncal = 10 x 20. 300. holhukal = 15 x 20. 400. hunbak = 1 tying around. 500. hotubak. 600. lahutubak 800. calbak = 2 x 400. 900. hotu yoxbak. 1000. lahuyoxbak. 1200. oxbak = 3 x 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 x 20. 50. ompoalli ommatlactli = 40-10. 60. eipoalli, or epoalli, = 3 x 20. 70. epoalli ommatlactli = 60-10. 80. nauhpoalli = 4 x 20. 90. nauhpoalli ommatlactli = 90-10. 100. macuilpoalli = 5 x 20. 120. chiquacempoalli = 6 x 20. 140. chicompoalli = 7 x 20. 160. chicuepoalli = 8 x 20. 180. chiconauhpoalli = 9 x 20. 200. matlacpoalli = 10 x 20. 220. matlactli oncempoalli = 11 x 20. 240. matlactli omompoalli = 12 x 20. 260. matlactli omeipoalli = 13 x 20. 280. matlactli onnauhpoalli = 14 x 20. 300. caxtolpoalli = 15 x 20. 320. caxtolli oncempoalli. 399. caxtolli onnauhpoalli ipan caxtolli onnaui = 19 x 20 + 19. 400. centzontli = 1 bunch of grass, or 1 tuft of hair. 800. ometzontli = 2 x 400. 1200. eitzontli = 3 x 400. 7600. caxtolli onnauhtzontli = 19 x 400. 8000. cenxiquipilli, or cexiquipilli. 160,000. cempoalxiquipilli = 20 x 8000. 3,200,000. centzonxiquipilli = 400 x 8000. 64,000,000. cempoaltzonxiquipilli = 20 x 400 x 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) x 400 x 800 + (15 + 4) x 20 x 8000 + (15 + 4) x 8000 + (15 + 4) x 400 + (15 + 4) x 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.

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