4.1 THE EXISTENCE AND NATURE OF 10-CYCLE SYSTEMS
Several types of 10-cycle systems have been documented in the literature on natural language numerals. The most common type is such that there are distinct words for the numbers 1 to 10; the second decade numbers 11 to 19 are additive compounds which usually take the form "10+n" (or "n+10") where "n" takes the values 1 to 9 respectively so that 11 = 10+1, 12 = 10+2, and so on; the decades from 20 to 90 are normally multiplicative compounds of the form "nx10", where "n" takes the values 2 to 9 respectively so that 20 = 2x10, 30 = 3x10, and so on. There is often a distinct word for "hundred" and the system may therefore possess a secondary 100-cycle. Such "pure" 10-cycle systems are found in Asia, the Middle East, Africa, Europe, and North America. Various Chinese languages and Japanese possess this type of system as do the Semitic languages. Most of the languages belonging to the Indo-European family possess 10-cycle system variants. Menninger, for example, lists the 10-cycle numerals of 30 daughter languages of Proto Indo-European which itself has a 10-cycle system. Many of these languages have systems with irregularities of various kinds and which deviate to some extent from the standard defined above. The Germanic languages, for example, have suppletive forms of 11 and 12. The Romance languages, in forming the compounds 11 to 19, have the peculiarity in which, up to 15 or 16, the units are placed before the "ten" morpheme, and thereafter the units are placed after it. More significant, however, is the appearance, in the counting systems of various languages, the remnants of a 20-cycle: French, for example, has 80 = 4x20 and 90 = (4x20)+10; Breton has 40 = 2x20, 60 = 3x20, and 70 = 10+ (3x20). The non-Indo-European language, Basque, has a counting system with a fully formed (10, 20) cyclic pattern which Schmidt says is also found "in the Munda languages of India and in the Tibeto-Chinese groups of the Himalayas, in Nicobarese, in the north (and south) Caucasian languages".
In the North American Indian languages, Eells reports that the majority of the counting systems in his survey were "decimal systems", that is having a 10-cycle, and, more rarely, "decimal-vigesimal", that is having a (10, 20) cyclic pattern. Hymes, in her analysis of the Athapaskan Indian numeral systems, distinguishes several variants of the 10-cycle system: (a) the "decimal", with distinct terms for 1 to 10; (b) the "decimal-pairing", with distinct terms for 1 to 7, and 10, and with 8 and 9 having the "pairing" structure 8 = 2x4 and 9 = (2x4)+1; (c) the "decimal-subtractive", in which there are distinct terms for 1 to 7, and 8 and 9 contain 2 and 1 respectively, plus a morpheme or group of morphemes signifying roughly "it lacks". Hymes notes that "No pure subtractive system (6 containing 4, 7 containing 3, and 8 and 9 containing 2 and 1) has been found among the Athapaskan languages, but it exists elsewhere." Similarly, Dixon and Kroeber, in their discussion of Californian (Indian) numeral systems, indicate the existence of several variant 10-cycle systems: "A few Californian languages show a decimal system throughout, even to being based on hundreds from one hundred up ... Sometimes a decimal system changes above twenty to a vigesimal one, for which an analogy is not far distant in French." Dixon and Kroeber also note the use of the "duplicative" method of forming compound numerals (cf. Hymes' "pairing") in which 6 has an implied 2x3 structure and 8 is 2x4.
One notable system, which has several of the features mentioned above, is that of the Ainu people who at one time were located in Hokkaido, Sakhalin Island, and the Kurile Islands. The data, which derive from Laufer and from Peng and Brainerd, indicate that the Ainu numerals are such that there are distinct terms for 1 to 5, and 10, and that the numerals from 6 to 9 are formed by subtraction from 10 so that, for example, 6 has an implied construction 10-4, 7 = 10-3, 8 = 10-2, and 9 = 10-1. Laufer notes that "the numbers from eleven to nineteen are formed on the scheme 1+10, 2+10, ... The unit of all higher counting is represented by the figure 20." Thus 40 = 2x20, 60 = 3x20, 80 = 4x20 and 100 = 5x20. The odd decades are formed subtractively so that, for example, 30 is "ten less than two twenties." The system thus possesses elements of a 10-cycle system and of a 20-cycle system together with with the subtractive formation, or what Menninger calls "back-counting", for the numerals 6 to 9, and the odd decades. The use of the subtractive construction for certain compound numerals is not uncommon. Armstrong says of the Nigerian Yoruba that "one of the most characteristic features of Yoruba numerals is the use of subtractive numerals in the higher decades ... The compound word eedogun says literally 'five reduces twenty'. Mathematically, it means 20-5. Similarly, 'sixteen' is erin din logun, which means literally, 'four of a reduction from twenty', or 20-4. 'Seventeen', 'eighteen', and 'nineteen' are formed similarly. This pattern is preserved almost up to two hundred. The early members of a decade are formed by addition; the last five members are formed by subtraction." Conant also gives examples of the subtractive construction used in counting systems in various parts of the world, notably that of the spoken forms in Classical Latin of the numerals 18 and 19 which are, respectively, duodeveginti and undeveginti, that is 2 from 20 and 1 from 20.
One final example of what is essentially a 10-cycle system and which is relevant to our subsequent discussion is the numeral system of the Babylonian Sumerians. For the first few decades this system possesses an irregular (10, 20) cyclic pattern so that 30 = 3x10, 40 = 2x20, and 50 = (2x20)+10. However, at 60, a new distinct word is introduced ("the great unit") and thereafter we have an additional 60-cycle so that 120 = 60x2, 180 = 60x3, and 600 = 60x10. Thureau-Dangin notes, however, that "the whole system of Sumerian numeration which is, properly speaking, not a sexagesimal system, but a system built up on two alternative bases of 10 and 6."
In summary then, the literature reveals a number of important variants of the 10-cycle numeral system. In some of these, there are unusual methods of compounding the numerals 6 to 9 (I omit, of course, the 5-cycle construction of these numerals which have been dealt with in Chapter 3). One method of compounding is the subtractive one in which we have at least 8 = 10-2 and 9 = 10-1. In some, systems 7 and even 6 are also expressed subtractively. It should be noted that in these compounds the "ten" morpheme may not appear explicitly and 8, for example, may have the translation "it lacks two", or something similar. A further way of compounding the numerals 6 to 9 is the "pairing" method in which we may have, for example, 6 = 2x3 and 8 = 2x4. In such systems there may be a distinct word for 7 (and also for 9), however it is relatively common to find 7 = (2x3)+1 and 9 = (2x4)+1. In the construction of the numerals of the second decade, that is from 11 to 19, we usually find additive compounds of the form "10+n" or "n+10". Such a pattern may not persist for the whole decade: we find some systems which may have 11 to 15 expressed as, say, "n+10" and then 16 to 19 expressed in a different fashion, either "10+n" or "20-n". The construction of the decades 20 to 90 is usually multiplicative, that is 2 = 2x10 and 50 = 5x10, and so on. We do find, however, systems possessing secondary (and tertiary) cycles greater than 10 and less than 100, for example those systems with (10, 20) and (10, 20, 60) cyclic patterns. One other irregularity not uncommon among 10-cycle systems is what Hurford calls "base-suppletion" in which we have the numeral 10 differing from the "ten" morpheme used in compounds. As we shall see below, these 10-cycle variants are also present in the numeral systems of New Guinea and Oceania.
4.2 10-CYCLE SYSTEMS IN THE PAPUAN LANGUAGES
The occurrence of any of the 10-cycle variants among the NAN languages is very rare and indeed there are only 16 instances in our sample of 430, that is just under 4% of the total. Three types of 10-cycle system are distributed among the various phylic groups as given in Table 30 .
Distribution of 10-cycle Systems Among NAN Phylic Groups
(10, 100) (10, 20) (10, 20, 60)
Small Phyla - - -
West Papuan 1 2 -
East Papuan 8 - -
Torricelli - - -
Sepik-Ramu 1 - -
Trans-New Guinea 2 1 1
Totals 12 3 1
The three West Papuan languages with 10-cycle systems are Karon-Pantai, Moi, and Madik. Their numerals are given in Table 31.
Numerals of Three NAN Languages of the West Papuan Phylum Which Have 10-Cycle Systems
Karon-Pantai Moi Madik
1 dik mele, mere dik
2 we, uwe ali uwi, ue
3 kri, gri tolu(k) dili
4 at fat at
5 mek, wek mafu(t) mek
6 mat (t)anim mohemat
7 fit, mik fotu momit
8 mengwo, mko walu mosembu
9 mesi, misi (walu)si mosi
10 mesju, musiu sitit, utmere motu
20 musiu-we ut-ali, na-mere-igi moro
30 musiu-kri ut-toluk, fe-toluk moru-mutu-tongal
40 musiu-at ut-fat, najali-igi moru-uwi
Each of these is located on the Vogelkop in western Irian Jaya and it seems likely that their possession of 10-cycle systems has occurred as a result of the influence of neighbouring AN languages. Indeed Moi has clearly borrowed several AN numerals while Karon-Pantai and Madik both have several numerals (4, 7, and 9) which appear to have been influenced by an AN source. The inference that may be drawn is that the three NAN language groups have augmented their numeral lexis by borrowing from AN language sources and it may be, that by doing this, each language now has a 10-cycle system it did not originally possess.
The East Papuan Phylum languages show the greatest incidence of 10-cycle systems with a total of eight languages possessing them. Three of these, Anem, Buin and Yele, are located in islands east of the New Guinea mainland. Their numerals are given in Table 32 .
Numerals of Three NAN Languages of the East Papuan Phylum (PNG) Which Have 10-Cycle Systems
Yele Anem Buin
1 ngme udeta nonumoi
2 mio ngiak keitako
3 pyiile bik paigami
4 paadi tanol korigami
5 limi esi upugami
6 weni kamli tugigami
7 pyidu kisa paigami tuo
8 waali damil keitako tuo
9 tyu damli kampuro
10 y:a le kiburo
20 myo y:a le ngiak kikoko
The Yele numerals clearly show the effect of AN influence; this is not readily apparent for either the Buin or Anem numerals, nor indeed is there any obvious lexical influence to be detected in any of the remaining five numeral systems, all of which belong to languages spoken in the Solomon Islands. Three of these are given in Table 33 below. Each of the Solomon Islands' NAN languages, and Buin and Anem as well, possess AN neighbours and it seems likely that the AN influence has been such as to effect a change in the cyclic patterns of the NAN systems without direct lexical borrowing occurring. There is little uniformity between the various systems shown in Tables 32 and 33. While most of these have distinct numerals for 1 to 10 it can be seen that both Buin and Nanggu have subtractive constructions for at least the numerals 7 and 8 (which contain, respectively, the numerals 3 and 2). In this study, data have been acquired for 23 languages of the East Papuan Phylum. The majority (13) of these have 5-cycle numeral systems; the eight discussed here have 10-cycle systems, and two possess systems which have 2-cycle variants. It is, of course, impossible to know what the original numeral system of the East Papuan Phylum languages may have been. What seems likely, however, is that due to their removal from the influence of other NAN languages and to their relative isolation in areas which are predominantly AN, they have had their original systems changed in some way; for the nine languages discussed here the change has been such that they now possess 10-cycle systems.
Numerals of Three NAN Languages of the East Papuan Phylum (Solomon Islands) Which Have 10-Cycle Systems
Mbaniata Savosavo Nanggu
1 tufi ela töti
2 eri endo tüli
3 hie igiva tütü
4 avo agava tupwa
5 sondu ara mööpwm
6 tumbi pogoa temuu
7 ohio pogora tutüü
8 mbihio kui tumulii
9 mbovoho kuava tumatee
10 to atale napnu
20 eri to - napnu li
Three other NAN languages appear to possess 10-cycle systems. These are Wapi, of the Sepik-Ramu Phylum, several dialects of Enga, and Lembena, the latter two belonging to the Trans-New Guinea Phylum (see Table 34 for the numerals of these languages). Each of these languages is situated in the Enga Province (PNG). Both the Wapi and Lembena numeral sytems appear to have been strongly influenced by the dominant Enga language, the Wapi numerals, indeed, having been displaced by the Enga numerals. The original Enga counting system, to be discussed subsequently under 4-cycle systems, is a complex one with a (4, 60) cyclic pattern. This system has been modified by the process of truncation of the original numeral sequence at 10: from 1 to 9 the numerals are identical to the (4, 60) system; a new term akalita has been introduced for 10: this means "man" and thus 10 has the meaning "the fingers of one man". It is likely that the change in the Enga system affected both the Wapi and Lembena systems and that the adoption of a 10-cycle system in each of these languages is a relatively recent innovation.
Numerals of Three NAN Systems: Enga (Traditional), the Layapo Dialect of Enga, and Wapi
Enga (Traditional) Enga (Layapo d.) Wapi
1 me(n)dai mendai mendai
2 lapo lapo lapo
3 tepo tepo tepo
4 kitome(n)de kitumende kitumende
5 yungi kondape kingi yangi
6 tokage yangi mange -
7 kalage sakaita -
8 tukulapo tukulapo -
9 tukuteponya me(n)dai mange mendai wakitao -
10 tukuteponya lapo akalita akalita
11 tukuteponya tepo akalita kisa mendai -
12 tukuteponya gato akalita kisa lapo -
One final NAN 10-cycle system is considered here. This is the unusual and notable Ekagi (or Kapauku) system as shown in Table 35.
Numerals of the Ekagi Counting System
1 ena 11 enama (gati)
2 wia, wisa 12 wiama (gati)
3 wido 20 gatima (gati) or mepina
4 wi, wie 30 jokagati or amonato
5 idebi 40 mepija, mepiia
6 benomi 50 gati beu
7 pituo 60 bado or muto
8 waruwo 90 bado wado jokagati
9 ije, ise 120 muto wia
The Ekagi, located in the Wissel Lakes region of Irian Jaya, possess a counting system which has essentially a (10, 20, 60) cyclic pattern. We have, for example, 40 = 2x20, 50 has the subtractive construction "ten without" or 60 - 10. There is a distinct term for 60 and 120 = 60x2. There are two terms for 30: "child of ten" (jokagati) or "a half" (amonato). The cyclic structure of this system is clearly similar to that of the Babylonian Sumerians discussed earlier. Indeed this similarity prompted de Solla Price and Pospisil to suggest that perhaps the Ekagi system was "a survival of Babylonian arithmetic in New Guinea", a suggestion which subsequently caused the authors to be taken to task by Bowers and Lepi who deplored "the racist implications of Price and Pospisil's fanciful effort to derive Kapauku numeration from Babylonia." In Bowers' response to a defence of their position by Pospisil and de Solla Price she asserts that "too much store need not be placed on the 60-base of Kapauku numeration; body-count totals, sometimes functioning as bases, have been recorded for 19, 23, 25, 27, 33, 37, 44, 47, 68, etc., in NAN languages. The sexagesimal system is really not all that unique."
Bowers' point about the existence of unusual bases in New Guinea (not, however in Melanesia generally) is a valid one. Such bases, as she points out, are usually associated with body-part tally systems of the sort described in Chapter 2 but there is no evidence that the Ekagi system is or was a body-part one; indeed the data appear to indicate elements of a digit-tally system with both 10- and 20-cycles. The sexagesimal system is certainly not unique to the Babylonian Sumerians: two instances exist in New Guinea: the Ekagi itself and the Mae dialect of Enga, the latter possessing a (4, 60) cyclic pattern. While 60-cycle systems are therefore not unique to Babylonia, nevertheless the point needs to be made that this type of system is very unusual. Yet there is no need to posit the diffusionist view of a distant Babylonian source for the Ekagi system if it can be shown, as seems likely, that it was a local development. Bowers quotes Father Drabbe's view that the Ekagi system "was essentially senary", that is (6-cycle), "with the decimal aspects arising from Indonesian influence." In disagreeing with this view, Bowers notes that "there is no basis for positing an original 6-base in highland Irian Jaya counting systems." However 6-cycle systems do exist in languages situated on the southern coast of Irian Jaya and it may be the case that the Ekagi were influenced by the 6-cycle counters, as indeed they were influenced by a 10-cycle AN source. As Thureau-Dangin noted, in discussing the Babylonian system, the 60-cycle was a result of the hybrid of two bases: 10 and 6, and it may well be the case that the Ekagi system is, similarly, as Drabbe has suggested, a hybrid of a 6-cycle (NAN) system and a 10-cycle (AN) system. It would seem possible, then, that in this particular example, an independent inventionist view, rather than a diffusionist one, may account for the facts.
4.3 10-CYCLE SYSTEMS IN THE AN LANGUAGES OF NEW GUINEA
The reconstruction of the languages which are the ancestors to the AN languages spoken in New Guinea today has indicated that the numeral systems of both Proto Austronesian (PAN) and Proto Oceanic (POC) were 10-cycle systems. Ross, for example, has reconstructed the POC numerals which are given in Table 36.
The Numerals 1 to 10 of POC As Reconstructed By Ross
1 *kai, *sa, *tai
4 *pat, *pati
As we have seen in the previous chapters, many AN languages do not possess 10-cycle systems: 32 AN languages spoken in New Guinea now have variants of 2-cycle systems and 222 AN languages spoken in New Guinea and Island Melanesia have variants of 5-cycle systems. Indeed, of the 453 AN languages in our sample, there are only 187 which have 10-cycle systems, or 41% of the total. Some 92 of these are located in New Guinea region, 16 in Irian Jaya (all of which are non-Oceanic AN) and 76 in PNG and the north-west Solomon Islands (all of which are Oceanic AN). Table 37 shows the distribution of the 10-cycle systems among the various Oceanic AN clusters.
Distribution of 10-Cycle Systems Among the Oceanic AN Clusters
(10, 100) (10, 20)
North New Guinea Cluster 0 0
Papuan Tip Cluster 9 0
Admiralties Cluster/ St Matthias 23 0
Meso-Melanesian Cluster 41 3
Totals 73 3
Of the 78 languages altogether in the North New Guinea Cluster, none has a 10-cycle system: 52 possess a 5-cycle variant, 2 possess 4-cycle systems, and 24 possess a 2-cycle variant. In the Papuan Tip Cluster, of the 41 languages for which data were obtained, 7 possess a 2-cycle variant, 23 (and possibly a further 2) possess a 5-cycle variant, and only 9 have a 10-cycle variant. Of the 24 languages spoken in the Manus Province (and which belong to the Admiralties Cluster), 2 possess a 5-cycle variant and the remaining 22 have 10-cycle systems. The Meso-Melanesian Cluster comprises 20 languages of the North-West Solomonic Group, all of which have 10-cycle systems, and a further 44 languages within PNG. Of these, 24 possess 10-cycle systems while 18 possess a 5-cycle variant, and one language, Tomoip, has a 2-cycle variant. Thus, of the four clusters, the only ones which have retained something of the original POC 10-cycle nature for their numeral systems are the Admiralties and Meso-Melanesian Clusters.
The POC system, as shown in Table 36, possesses ten distinct terms for the numerals 1 to 10. This is not true for many of its daughter languages which often have compounds rather than distinct terms for the numerals in the second pentad. The main type of 10-cycle system, of which the POC system is one example, has ten distinct terms for 1 to 10, a further term for 100, and often a further term for 1000; such a system has a cyclic pattern of (10, 100, 1000). There are two major variants of this system. The first variant I have termed the "Manus" type, largely because this is found almost exclusively in the Manus Province (PNG). This type has distinct terms for the numerals 1 to 6, and 10 (and normally 100 as well). The numerals 7, 8, and 9 are, however, compounds which contain, respectively, the numerals 3, 2, and 1, that is the construction of the compounds is subtractive. In two cases the subtractive construction extends also to the numeral 6. Some examples of the "Manus" type of 10-cycle system are shown in Table 38, together with Ross's reconstructed numerals of Proto Eastern Admiralties (PEAd). Of the 22 languages of the Admiralties Cluster which possess 10-cycle systems, 21 are of the "Manus" type, the exception being Wuvulu-Aua, the westernmost language of the Cluster and which is discussed further below. The only other AN language to possess the "Manus" type of system, both in New Guinea and Oceania, is the Mioko dialect of Duke of York in the East New Britain Province (PNG).
Numeral Systems of the "Manus" Type Including Those of PEAd
Titan Papitalai Likum PEAd
1 esi ti esi *si-
2 eluo moruah rueh *ru-
3 etalo motalah taloh *tolu-
4 ea(h) mohahu hahu *fa-
5 elima molimah limeh *lima-
6 ewono mowonoh chohahu *ono-
7 andratalo moadotalah chotaloh *(a)nto-tolu
8 andraluo moadoruah chorueh *(a)nto-ru
9 andrasi moadoti choesi *(a)nto-si
10 eakou moasengul senoh
One other important feature of the languages of the Admiralties Cluster, which is discussed at greater length in Chapter 6, is the use of "numeral classification". This is found in all the languages of Manus Island and the islands peripheral to it. Essentially, numeral classification consists in this: the universe of countable objects is divided into (largely) discrete classes; in counting the objects of a particular class or in stating the cardinal number of objects of a particular class (in, for example, a noun phrase), use is made of the construction "numeral root + classifier" (rather than just a numeral). Thus one does not use the phrase, for example, "three spears" but rather "three-(classifier for long thin objects) spears". Some languages have in excess of 40 classes. There are usually numeral roots for 1 to 9; in certain languages, however, numeral classification is only used with the numeral roots 1 to 4, after which a general set of numerals is used. Data on the Gele' dialect of the language Ere-Lele-Gele'-Kuruti were collected by a medical officer and amateur linguist Dr W. E. Smythe in the 1940s who reported 43 classes and gave the "numeral root + classifier" constructions for each class for 1 to 9; this unpublished report  constitutes one of the few comprehensive records of a rapidly disappearing linguistic phenomenon in the Manus languages. Numeral classification is not a phenomenon found exclusively in the languages of the Admiralties Cluster: it is found in other AN language groups, both in PNG and Oceania, notably those of the Massim Group in the Papuan Tip Cluster as well as in the languages of Micronesia. It is also found in various NAN languages.
A second major variant, which I have termed the "Motu" type, is largely found in the AN languages of the coastal region to the east and west of Port Moresby in the Central Province (and National Capital District) of PNG. These languages are members of the western part of the "Peripheral" Papuan Tip Cluster. Taking the example of the numerals of Motu itself (as given below in Table 39), we have distinct terms for the numerals 1 to 5, 7, and 10. The numerals 6, 8, and 9 are each compounds formed by what has been termed above as the "pairing" or "duplicative" method, that is 6 = 2x3, 8 = 2x4, and 9 = (2x4)+1. The "Motu" type may have, as do the Nara, Gabadi, Roro, Keapara (generally), and Sinagoro (Balawaia dialect) systems, a compound for the numeral 7 such that 7 = (2x3)+1. There is also some evidence that one of the Keapara dialects has subtractive compounds for both 7 and 9 such that 7 = 8-1 and 9 = 10-1 while still retaining 6 = 2x3 and 8 = 2x4. Table 39 shows these "Motu" type variants.
Variants of the "Motu" Type of 10-Cycle System
Motu Nara Keapara
1 ta kaonamo ka
2 rua lua lualua
3 toi koi koikoi
4 hani vani vaivai
5 ima ima imaima
6 tauratoi kalakoi kaulakoi(koi)
7 hitu kalakoi ka mapere aulavaivai
8 taurahani kalavani kaulavaivai
9 taurahani ta kalavani ka mapere ka gahalana
10 gwauta ouka gahalana
While I have not included it as a defining feature of the "Motu" type, several systems of that type possess, using Hurford's term, "base-suppletion" in which the word for the numeral 10 is not the same as the "ten" morpheme used in multiplicative compounds. Two examples, those of Nara and Motu, are given in Table 40.
Examples of Base-Suppletion in the "Motu" Type Numeral Systems
10 gwauta ouka
11 gwauta ta ouka ka
20 ruahui lua na vui
30 toi ahui koi na vui
40 hani ahui vani na vui
The "Motu" type of numeral system is not found outside the Central Province (PNG). Two systems which have elements of the "Motu" type, however, are Wuvulu-Aua, of the Admiralties Cluster, and Marshallese, an Oceanic AN language found in Micronesia. These are given in Table 41.
Two Numeral Systems With Elements of the "Motu" Type
1 ai, aiai juon
2 guai ruo
3 oduai jilu
4 guineroa emen
5 aipan lalim
6 oderoa (3x2) jiljino (3+3)
7 oderomiai (3x2 + 1) jiljilimjuon (3+3+1)
8 vaineroa (4x2) rualitök ("give two", i.e. to make 10)
9 vaineromiai (4x2 + 1) ruatimjuon (8+1?)
10 vapa ani jongoul
The "Manus" and "Motu" types of 10-cycle variant discussed above occur in the Admiralties and Papuan Tip Clusters. The 44 10-cycle systems which are found in the languages of the Meso-Melanesian Cluster (with the exception of the Mioko dialect of Duke of York) are all normal 10-cycle types without compounds for the numerals of the second pentad. These systems, most of which have numerals which are clearly reflexes of the POC numerals, are found mainly in the West New Britain, New Ireland, and North Solomons Provinces (PNG), as well as the north-west part of the Solomon Islands. It is not uncommon, particularly for the Solomon Islands languages, for the numerals 1 to 9 to have a prefix (in some cases this may be a fossilized classifier). Three examples of systems belonging to languages of the Meso-Melanesian Cluster are given in Table 42 below.
Numerals of Standard 10-Cycle Systems of Three Languages of the Meso-Melanesian Cluster
Nakanai Kandas Mbareke
1 isasa takai meka
2 ilua aru karua
3 itolu utul hike
4 ivaa uvat kambuto
5 ilima tilim kalima
6 iwolo vonom kaonomo
7 ivitu mavit kanjuapa
8 iwalu tival kavesu
9 walasiu lisu kasia
10 savulu sa singino nangguru puta
None of the five Oceanic AN languages situated in Irian Jaya possesses a 10-cycle system. Of the 33 non-Oceanic AN languages, however, 16 possess 10-cycle systems: 8 of these have (10, 100) cyclic patterns and 8 have systems with (10, 20) cyclic patterns and indeed these constitute about half of all the (10, 20) systems found in the 453 AN languages in our sample. At least 6 occur in the Polynesian languages and a further 3 occur in the North-West Solomonic group of the Meso-Melanesian Cluster. The geographical distributions of both types of 10-cycle system for the NAN and AN languages in the New Guinea area are shown in Map 12.
4.4 10-CYCLE SYSTEMS IN ISLAND MELANESIA
The distribution of 10-cycle systems in the languages of Island Melanesia (excluding the Polynesian Outliers which are discussed subsequently) is given in Table 43 below. The number of AN languages in the Solomon Islands which possess 10-cycle systems is 51, however 23 of these belong to the North-West Solomonic group and have been included in our discussion above of the Melanesian Cluster. (The distribution of 10-cycle systems in the Solomons is shown in Map 13).
Distribution of 10-Cycle Systems in Island Melanesia
Solomon Islands 28 (51)
New Caledonia 0
Total 50 (73)
Of the remaining 28 languages in the southern part of the Solomon Islands, all of which possess systems with (10, 100) cyclic patterns, 22 belong to the South-East Solomonic group and 6 belong to the Eastern Outer Islands. In Vanuatu, 19 of a total of 102 (non-Polynesian) Oceanic AN languages possess 10-cycle systems (see Map 14), the remaining 83 possessing systems with 5-cycle variants. In New Caledonia, none of the 27 Oceanic AN languages possesses a 10-cycle system; each of these has instead a 5-cycle variant. In Fiji, the numeral systems of the Eastern and Western dialect chains of Fijian each has a (10, 100) cyclic pattern as does that of Rotuma. None of the 10-cycle systems of this region are of the "Manus" or "Motu" types; they conform rather to the standard 10-cycle type as shown in Table 44 where three examples are given: Nggela (South-East Solomonic), Valpei (North-Central Vanuatu), and the Mbau dialect of Fijian. 
The Numerals 1 to 10 for Three 10-Cycle Systems From Island Melanesia
Nggela Valpei Fijian
1 sakai tewa ndua
2 rua rua rua
3 tolu tolu tolu
4 vati vati va
5 lima lima lima
6 ono ono ono
7 vitu pitu vitu
8 alu alu walu
9 hiua tchiwa thive
10 hangavulu sanawulu tini
4.5 10-CYCLE SYSTEMS IN POLYNESIA
Data were acquired in this study for 35 of a possible 36 Polynesian (PN) languages. These comprise: (a) the 14 PN Outliers which, with the exception of two located in Micronesia, are scattered throughout Island Melanesia, and (b) 22 languages located in "Triangle" Polynesia. Of the 35 PN languages, only one, Faga-Uvea, spoken in the Loyalty Islands of New Caledonia, does not have a 10-cycle numeral system. Faga-Uvea, one of the Outliers, possesses a 5-cycle variant. The data on the PN languages are such that it is not possible in every case to distinguish whether a numeral system has a (10, 100) or a (10, 20) cyclic pattern. These systems, however, are remarkably homogeneous in character and do not exhibit the diversity so apparent in the AN numeral systems of New Guinea and Island Melanesia. In Table 45, the reconstructed numerals of Proto Polynesian (PPN) are given, together with those of Takuu (an Outlier located in PNG), Samoan and New Zealand Maori (both located in Triangle Polynesia).
Numerals 1 to 10 for PPN and Three PN Languages
PPN Takuu Samoan N. Z. Maori
1 *ha, *taha tasi tasi tahi
2 *rua lua lua rua
3 *tolu tolu tolu toru
4 *faa fa'a faa whaa
5 *lima rima lima rima
6 *ono ono ono ono
7 *fitu fitu fitu whitu
8 *walu varu valu waru
9 *hiwa sivo iva iwa
10 *hangafulu sinahuru sefulu ngahuru
Base-suppletion is not uncommon in the PN numeral systems. It is often the case that the numeral 10 is used to construct the compounds 11 to 19 but, for the decades 20 (or 30) to 90, a different "ten" morpheme is used. We have, for example, Takuu, Luangiua and Hawai'i use the constructions as given in Table 46.
Examples of Base-Suppletion in Three PN Languages
Takuu Luangiua Hawai'i
10 sinahuru sengahulu 'umi
11 sinahuru ma tasi sengahulu na kahi 'umi kuma kahi
20 mata rua kipu lua (iwakalua)
30 mata toru kipu kolu kana kolu
40 mata fa - kana ha
One possible explanation for this phenomenon is that the decades of the systems for the languages shown, as well as for some other PN languages, were constructed using numeral classification, i.e the irregular "ten" morphemes are in fact "decade" or "group of ten" classifiers to which a numeral root is suffixed. For example, in Kapingamarangi, one of the PN Outliers located in Micronesia, the numeral 10 is ehoru, but there exists a "group of ten" classifier mada- to which the numeral roots for 2 to 9 are suffixed in order to form the decades 20 to 90. Thus, 20 is mada-lua, (30 is irregular: mo-tolu), 40 is mada-ha, 50 is mada-lima, and so on. Similarly, there exist "group of ten" classifiers in Nukuoru (mada-), Nuguria and Nukumanu (both tipu-), Takuu, Tikopian and Sikaiana (all mata-), Luangiua (kipu-), and Hawai'i (kana-). While this method of constructing decades is not uncommon among the Outliers, it is relatively uncommon among the Triangle languages which do not generally exhibit base-suppletion and in which the numeral 10 does appear in the decades 20 to 90 which are multiplicative compounds. Thus in Tongan, for example, we have 10 = hongo-fulu, 20 = uo-fulu, 30 = tolungo-fulu, and 40 = fango-fulu, and so on, where -fulu is the 10 morpheme.
A number of (mainly) Triangle PN languages have numeral sytems which possess a distinct term for 20 and which have a secondary 20-cycle. This term, tekau and its reflexes, often has the sense "ten pairs", the phenomenon of counting objects in twos and fours being common throughout most of Polynesia. In New Zealand Maori, tekau is recorded in some vocabularies as 10 and in others as 20. In Tongan, tekau is used for 10 but only in counting pairs; in Tongareva, tekau is a numeral classifier used when counting coconuts. At least 8 and possibly 9 Triangle languages have tekau (takau, ta'au) for 20 (or 10 pairs) and at least 6 of these have systems with (10, 20) cyclic patterns. These are Minihiki-Rakahanga, Tahitian, New Zealand Maori, North Marquesan, Pa'umotu , and Mangareva.
4.6 10-CYCLE SYSTEMS IN MICRONESIA
Each of 13 Micronesian (MC) languages for which data have been obtained possesses a 10-cycle numeral system. Of these languages, 11 exhibit the phenomenon of numeral classification while two, Marshallese and Kosraean, have largely lost it. In the enumeration of objects of a particular class each language makes use of the construction "numeral root + classifier", the same construction, in fact, as that found in the languages of the Admiralties Cluster and exactly the opposite of that found in the PN languages. The classifiers are suffixed to the numeral roots for 1 to 9. Each of the 11 languages has quantitative classifiers, or countable bases, for enumerating tens, hundreds, and thousands. There is, typically, a "tens" classifier which is suffixed to the numeral roots for 1 to 9 in order to form the decades 10 to 90. The systems are thus all 10-cycle ones (including those of Marshallese and Kosraean) and in fact have (10, 100, 1000) cyclic patterns. Examples of the classifier method of enumerating decades, hundreds, and thousands are given for three MC languages in Table 47.
Numeral Classifier Constructions for Tens, Hundreds, and Thousands for Three MC Languages
Kiribatese Woleaian Mokilese
10 te-bwi se-g ei-jek
20 ua-bwi reu-g riei-jek
30 ten-bwi seli-g jilih-jek
100 te-bubua sa-pagui e-pwki
200 ua-bubua ru-pagui rie-pwki
300 ten-bubua selu-pagui jili-pwki
1000 te-nga so-ngeras kid
2000 ua-nga ru-ngeras ria-kid
3000 ten-nga seli-ngeras jil-kid
Numeral classification is normally displayed in noun phrases; when "serial counting" of a set of objects takes place, each language possesses a special set of numerals which are not suffixed with a classifier. Several examples of serial counting numerals are given in Table 48. Serial counting is also carried out in other (non-MC) languages which possess numeral classification. In these cases we have numeral roots suffixed by a classifier which indicates that the objects being counted belong to a "general class" and these are used for serial counting rather than having a special set of unsuffixed numerals. Each of the MC languages also possesses a "general" class but has, in addition, the numerals used just for serial counting.
Numerals Used for Serial Counting in Three MC Languages
Ulithian Woleaian Trukese
1 yood yet, yut eet
2 ruy riew, ru uruuw
3 yeel yel een
4 faag fang faan
5 liim lim niim
6 wòòm wol, wul woon
7 fiis fis fuus
8 waal wal waan
9 diiw tiw, tiu ttiiw
10 se-yexe seg engoon
It is not certain whether serial counting is an innovation of the MC languages. Harrison and Jackson note that Codrington reports the "existence of serial counting in several languages of the Solomons. Such systems are also found in Roviana and Rotuman."
4.7 SUMMARY OF 10-CYCLE SYSTEM DATA
10-cycle numeral systems exist in New Guinea and Oceania. By far the majority (182) of these systems have a (10, 100) cyclic pattern while 20 have (10, 20) cyclic patterns. The distribution of these types among the AN and NAN language groups is as follows:
Distribution of 10-Cycle Systems Among the AN and NAN Languages of New Guinea and Oceania
(10, 100) (10, 20) (10, 20, 60)
NAN (N = 430) 12 3 1
AN (N = 453) 170 17 0
Totals 182 20 1
I have suggested that in the case of the 16 NAN languages it seems likely that the possession of a 10-cycle system is an innovation. That is, these languages, or languages from which these are descended, did not originally possess 10-cycle systems and that as a result of the influence of languages which do possess 10-cycle systems (AN or tok pisin or English) the NAN groups augmented their original systems, in some cases by lexical borrowing, to form systems with a 10-cycle. Eight of the systems belong to the NAN languages of the East Papuan Phylum which are situated in islands east of the New Guinea mainland and which are occupied by predominantly AN-speaking groups. Three NAN languages, spoken in the Enga Province (PNG) and which have 10-cycle systems, have probably acquired these not as a result of AN influence but rather as a result of the recent influence of the 10-cycle systems of tok pisin or English.
The situation with the AN languages is complex. On the one hand we have the results of comparative linguistics in which the reconstructed languages of Proto Austronesian and its daughter Proto Oceanic both possess 10-cycle numeral systems. We might thus reasonably expect all 453 of the AN languages considered here (which includes 33 non-Oceanic AN and 420 Oceanic AN languages) to possess 10-cycle systems as well. On the other hand, the evidence indicates that this is not the case and that only 187 of these AN daughter languages now possess 10-cycle systems. The remainder have either a 2-cycle variant (32), a 5-cycle variant (222), or a 4-cycle system (4), with some 8 languages not having a definite classification. None of the AN languages of the North New Guinea Cluster or those of New Caledonia has a 10-cycle system; the large majority of the languages of the Papuan Tip Cluster and of Vanuatu similarly do not possess 10-cycle systems. The majority of the 187 languages which do have 10-cycle systems belong to the Admiralties and Meso-Melanesian Clusters, the South-East Solomonic group, and the Polynesian and Micronesian groups, none of which is located on the New Guinea mainland. Those AN languages situated on or near the mainland and which have 10-cycle systems comprise 7 Papuan Tip languages (all of the "Motu" type) and 16 non-Oceanic languages which are situated in coastal or island Irian Jaya.
Several variants of the (10, 100) type of system have been delineated. The first of these I have termed the "Manus" type which is characterized by having a subtractive construction for the numerals 7, 8, and 9 (and sometimes 6 as well). This type of system is largely found in the AN languages of the Admiralties Cluster which are located on Manus Island and the small islands proximate to it. The exceptions are the Mioko dialect of the AN Duke of York language and also Buin and Nanggu, both belonging to the NAN East Papuan Phylum. The second important variant is the "Motu" type characterized by the pairing or duplicative construction such that 6 = 2x3, 8 = 2x4, 9 = (2x4) + 1 (and sometimes 7 = (2x3) + 1). This type is largely found in the AN languages located east and west of Port Moresby which belong to the Papuan Tip Cluster. These two variants, the "Manus" and "Motu" types, with their unusual second pentad constructions, both deviate from the standard POC numeral system which has distinct numerals for 6 to 9. Both types of system would thus appear to be localized innovations.
Generally speaking, the 10-cycle systems discussed here have a "10+n" construction for the numerals 11 to 19, where n takes, respectively, the values 1 to 9. One exception is the language Kuot, the only NAN language spoken in the predominantly AN-speaking New Ireland Province (PNG). This has, unusually, a "10+5+n" construction for the numerals 16 to 19 (where n takes the values 1 to 4 respectively) even though the numerals 6 to 9 do not have a 5-cycle construction. The construction of the decades for many of the 10-cycle systems is of the form "10xn" or "nx10". There are some important exceptions to this which exhibit what Hurford terms "base-suppletion" in which the "ten" morpheme used in constructing the decades 20 to 90 is not the same as the morpheme for 10. This is true particularly of those languages which possess, or show remnants of, numeral classification and which have "decade" or "group of ten" classifiers which are affixed with a numeral root.
Map 12. Distribution of 10-cycle systems (New Guinea)
Map 13. Distribution of 10-cycle systems (Solomon Is.)
Map 14. Distribution of 10-cycle systems (Vanuatu_
 Menninger (1969, p. 74).
 Armstrong (1962, p. 6).
 Conant (1896, p. 44).
 Menninger (1969, p. 74).
 Thureau-Dangin (1939, p.104).
 Hurford (1987, pp. 56-57).
 The data for Karon-Pantai may be found in DV5T9, Moi in DV5T5, and Madik in DV5T7.
 Anem is in the West New Britain Province (PNG): see AV4T41. The data on Buin, which is in the North Solomons Province (PNG), may be found in AV3T28. Yele is spoken in the Milne Bay Province (PNG) and the data are given in AV6T45,46.
 Compare, for example, the numerals 5 to 9 with the corresponding Proto Oceanic numerals shown in Table 36.
 The five NAN languages spoken in the Solomon Islands and which have 10-cycle systems are: Nanggu (see DV1T65), Mbilua (see DV1T60), Mbaniata (see DV1T61), Lavukaleve (see DV1T62), Savosavo (see DV1T63). Kazukuru , a further NAN language is now extinct:see DV1T59).
 The data for the Enga dialects are found in BV9T4,5,6,7. Those for Lembena are in BV9T8 and those for Wapi in BV9T9.
 The complete data on Ekagi are found in DV5T95,96. Further commentary may be found in de Solla Price & Pospisil (1966).
 de Solla Price & Pospisil (1966, p. 30).
 Bowers & Lepi (1975, p. 322).
 Pospisil, L. & de Solla Price, D. (1976). Letter to the Editor. Journal of the Polynesian Society, 85, 382-383.
 Bowers (1977, p. 112)
 Bowers (1977, p. 113).
 Bowers (1977, p. 113).
 Ross (1988, pp. 459-464).
 The two Manus languages which have subtractive constructions for 6, 7, 8, and 9 are Likum and the Levei dialect of Levei-Tulu. The data for these are located in AV2T20 and AV2T19 respectively.
 The Titan data are found in AV2T7. The Papitalai data are given in AV2T9 and the Likum data in AV2T20. The reconstructions of the Proto Eastern Admiralties (PEAd) numerals is given in Ross (1988, p. 344).
 Smythe's data are given in (AV2, pp. 23 - 34).
 The Massim Group languages which have numeral classification are: Kilivila (AV6T34,35), Muyuw (AV6T36,37), Budibud (AV6T38), Misima (AV6T30, 40), Nimowa (AV6T41,42), and Sud-Est (AV6T43,44).
 Data on the Micronesian languages, their numeral systems and numeral classification, are found in DV6.
 For example, Nasioi, Siwai, Buin, and Nagovisi are four NAN languages spoken in the North Solomons Province (PNG) which have numeral classification. The data for these are located in AV3.
 The languages which possess the "Motu" type of counting system and which are situated in the Central Province are: Nara, Doura, Roro, Gabadi, Motu, Sinagoro, and Keapara. All of these belong to Ross's Central Papuan Branch of the Papuan Tip Cluster.
 The Motu data are found in BV7T22, the Nara data are in BV7T4, and the Keapara data appear in BV7T24, 25.
 The Wuvulu-Aua data appear in AV2T27 and the Marshallese data are given in DV6T31,32.
 The Nakanai data appear in AV4T17. Those for Kandas are in AV1T19, and those for Mbareke are in DV1T15.
 The 8 non-Oceanic AN languages of Irian Jaya which possess (10, 100) systems are: Biak, Maya, Laganyan, Maden, Ron, Sekar, Arguni and Uruangnirin. The 8 languages which possess (10, 20) systems are: Onin, Bedoanas, Woi, Pom, Marau, Munggui, Ambai and Wabo. The data for all of these may be found in DV5.
 The 3 Meso-Melanesian languages which have (10, 20) systems are: Central-East Choiseul, Zazao, and Nggao. The data for these are given in DV1.
 The data on the South-East Solomonic Group and the Eastern Outer Islands Group are found in DV1. The cyclic patterns of these languages are given in DV1T71, 72.
 The cyclic patterns of the Vanuatu numeral systems are given in DV2T127-130.
 The cyclic patterns of the systems of New Caledonia are given in DV3T46.
 The cyclic patterns of Fijian and Rotuman are given in DV3T46.
 The Nggela data are given in DV1T25. The Valpei data appear in DV2T41 and the Fijian data are in DV3T40.
 The data on the PN languages are found in DV6.
 The data for Takuu may be found in AV3T6. The data for Samoan are given in DV4T23 and those for New Zealand Maori in DV4T29-31. The data on the Proto Polynesian numerals, located in DV4, derive from Pawley (1972, pp. 52-54).
 The data for Luangiua are in DV4T6,7 and those for Hawai'i are in DV4T39,40.
 The Kapingamarangi data are given in DV4T26,27.
 The Tongan data are given in DV4T26,27.
 The data for Kiribatese are found in DV6T33-38. Those for Woleaian in DV6T6-11, and those for Mokilese in DV6T26-28.
 A more complete table exhibiting the serial counting numerals is DV6T47.
 Harrison & Jackson (1984, p. 72).