CHAPTER 3

5-CYCLE SYSTEMS

3.1 THE EXISTENCE AND NATURE OF 5-CYCLE SYSTEMS

Edward Tylor, Professor of Anthropology at the University of Oxford, wrote in 1871 that "the general framework of numeration stands throughout the world as an abiding monument to primaeval culture. This framework, the all but universal scheme of reckoning by fives, tens, and twenties shows that ... the practice of counting on fingers and toes lies at the foundation of our arithmetical science."[1] As the data on counting systems of newly discovered indigenous societies became available during the nineteenth century, Tylor, like many of his contemporaries, was struck by the ubiquitous nature of the "quinary", or 5-cycle, systems. The dictum of the Brothers Grimm that "all numerals are derived from fingers"[2] appeared to hold true at least for the numeral "five" which, for a great many languages, was either identical to, or cognate with, the word for "hand". Tylor continued: "To count the fingers of one hand up to 5, and then go on with a second five, is a notation by fives, or, as it is called a quinary notation. To count by the use of both hands to 10, and thence to reckon by tens, is a decimal notation. To go by hands and feet to 20, and thence to reckon by twenties, is a vigesimal notation ... In surveying the languages of the world at large ... there prevails, with scarcely an exception, a method founded on hand counting, quinary, decimal, vigesimal, or combined of these".[3] It was commonly noted, too, that for the last type of system, the numeral 20 often had the meaning "one man", i.e. the fingers and toes of one man.

We thus find several types of 5-cycle numeral system reported in the literature. Schmidt,[4] for example, discusses three variants: the "pure" quinary, the "quinary vigesimal", and the "quinary decimal". The first corresponds, in Salzmann's terms, to a 5-cycle system which does not possess a secondary cycle and which has the symbolic representation: 1, 2, 3, 4, 5, 5+1, 5+2, 5+3, 5+4, 2x5, (2x5)+1, ... , 3x5, and so on. The "quinary vigesimal" system has a (5, 20) cyclic pattern which is such that it is "pure" quinary up to 20 after which a secondary 20-cycle comes into play and we have, for example, 40 = 2x20 and 100 = 5x20. A third type, the "quinary decimal", also referred to as the "imperfect decimal",[5] has a (5, 10) cyclic pattern so that the symbolic representation is: 1, 2, 3, 4, 5, 5+1, 5+2, 5+3, 5+4, 10, 10+1, ... , 2x10, and so on. A further type, discussed by Eells in his study of North American Indian counting systems,[6] is the "quinary-decimal-vigesimal" system, or in Salzmann's terms, one with a (5, 10, 20) cyclic pattern. Each of the four types mentioned here has a number of variants.

The 5-cycle systems are distributed throughout all the major continents. Schmidt's language atlas, associated with his Die Sprachfamilien und Sprachenkreise der Erde,[7] contains a counting systems map which shows the distribution of two types of "quinary" system throughout the world. The (5, 20) type occurs "sporadically in almost all the Australian linguistic groups; in nearly all the NAN languages of the (north) east coast and exterior of New Guinea, in the oldest Melanesian languages of New Caledonia, the Loyalty islands, etc".[8] In the Australian languages, the literature indicates that the (5, 20) system is usually associated with a variant of the 2-cycle numeral system. Dawson,[9] for example, provides data on two language groups found in western Victoria, each of which has a system which may be represented as follows: 1, 2, 3, 2+2, 5, 5+1, 5+2, ... , 2x5, (2x5)+1, ... , 20, and so on. Among Kluge's numeral lists[10] there is ample evidence of similar systems in other Australian languages while Harris[11] provides data on the existence of (5, 20) systems without the primary 2-cycle. The (5, 20) system is also found in the Americas. As Nykl noted in his 1926 article on quinary systems, "on North American soil, as far as is known, the quinary-vigesimal system does not exist east of the Rocky Mountains with but one exception, while it can be found everywhere along the whole Pacific Coast from Alaska to Southern California, and from there it becomes almost universal as far as Brazil."[12] Dixon and Kroeber, in their 1907 article on the counting systems of California, also note that for North America "as contrasted with the wide extension of thorough decimal systems, quinary-vigesimal systems occur but rarely. Outside of Mexico, they are to be found only among the Caddoan tribes,[13] the Eskimo, and in parts of California".[14] Schmidl's article on African counting systems indicates the existence of (5, 20) systems which are found in the Ivory Coast and central African regions. The existence of "quinary" systems in Africa is also confirmed by data in Conant[15] and in Pott who published, in 1847, a monograph devoted entirely to the quinary and vigesimal counting systems, Die quinare und vigesimale Zählmethode bei Völkern aller Weltheile.

The (5, 10) system, according to Schmidt's counting systems map, is found throughout Africa, Southeast Asia and North America. Schmidt notes that "in Australia and in the NAN languages it is found isolated and in rudimentary form; with the Melanesians it is fully displayed. In Asia it originally dominated the two great families of Austroasiatic and the Tibeto-Chinese languages ... So too, in North America; but in neither Mexico, Central America, nor in South America has the system spread widely."[16] Schmidt's definition of the (5, 10) system is that "the numbers of the second pentad are formed by composition with 'five' (6 = 5+1, 7 = 5+2, etc.) or by the pair system (especially 6 = 3x2, 8 = 4x2) or by subtraction (especially 9 = 10-1)".[17] This is a broader definition than will be adopted here and indeed it is one which Seidenberg has criticized.[18] Specifically, I will not regard as 5-cycle systems those which form the second pentad subtractively (as is found among the languages of the Manus Province in PNG or the Ainu language of Japan), or by doubling, as given by Schmidt for 6 and 8, and which is found in the Motu and related languages of the Central Province of PNG. By adopting this more stringent definition, the distribution of the (5, 10) system as given by Schmidt will no doubt be diminished, nevertheless the evidence from other sources indicates that the system is relatively widespread. Conant, for example, provides many instances of (5, 10) systems: some 35 from various African languages, 30-odd from North America, and several from Oceania.[19]

The (5, 10, 20) system, rarely mentioned in the literature usually because it is not distinguished from either the (5, 10) or the (5, 20) systems, is however mentioned by Eells who notes that for North American Indian languages "several systems show a combination of three digital bases in their formation."[20] The normal structure for this type of system is: 1, 2, 3, 4, 5, 5+1, 5+2, 5+3, 5+4, 10, 10+1, 10+2, ... , 10+5, 10+5+1, ... , 20, 20+1, ... , 20+10, ... , 2x20, and so on. Eells mentions a variant in which the quinary nature of the system does not become apparent until the second decade. The Welsh counting system is, in fact, of this type and is such that it has ten distinct numerals and then 11 to 19 are constructed as follows: 1+10, 2+10, 3+10, 4+10, 5+10, 1+5+10, 2+5+10, 3+5+10, 4+5+10. There is a distinct numeral 20 and we have 30 = 20+10, 40 = 2x20, 50 = 10+(2x20), and 60 = 3x20.

The three types of system discussed above are each important variants of 5-cycle systems. A fourth type, the "pure quinary" system (without a secondary cycle), Schmidt says "is found only in Saraweka, a South American Arowak language. Everywhere else it is combined either with the decimal or the vigesimal system."[21] There is also the possibility, in the absence of there being a sufficient amount of data, that some systems have been classified as "pure" quinary whereas they may in fact have a secondary or even tertiary cycle.

3.2 5-CYCLE SYSTEMS IN NEW GUINEA AND OCEANIA

Commenting, in 1907, on numeration in British New Guinea, Sidney Ray contrasted the counting systems of the NAN languages and those of the "Melanesian", or Austronesian, languages. The former, he said, "very rarely advances beyond five, and that as a rule only two, or at most three numerals are named ... In the Melanesian languages without exception, numbers can be named at least as far as five, and counting can be performed beyond, by fives, tens, or twenties".[22] Ray also noted that, for the AN languages, "counting is performed with the fingers, and in some the toes also are counted."[23] Similarly, Codrington, in 1885, commented on numeration in other parts of Island Melanesia and observed that "three systems of numeration which are based on the counting on the fingers are found ... In some of the islands of the New Hebrides group and in the Banks Islands the notation is quinary; in other islands of the New Hebrides, in Fiji and in the Solomon Islands, it is decimal; in the Loyalty Islands, New Caledonia and Anaiteum, the notation is, or was, vigesimal".[24] Thus the quinary nature of the counting systems of some AN language groups, and the widespread occurrence of such systems throughout New Guinea and Island Melanesia, has long been observed. As we shall see, Ray's comments about the non-quinary nature of the NAN languages do not generally hold true and that the various 5-cycle systems are found in AN and NAN languages alike. In the sections that follow, the main types of 5-cycle system mentioned above will be discussed. Each type has several variants in the way the second pentad is constructed and these will be dealt with as well.

3.3 SYSTEMS WITH A (5) or (5, 20) CYCLIC PATTERN.

The "pure" 5-cycle system is a rare phenomenon in New Guinea and Oceania: for almost every case, where sufficient data are available, there is evidence of a secondary cycle. One exception appears to be Kaulong, spoken in the West New Britain Province (PNG)[25] and which is such that the numeral 20 has an implied "5x4" construction:

Table 13

A Possible Example of a "Pure" 5-Cycle System: Kaulong in West New Britain (PNG)

_____________________________________________________________________

Kaulong

______________________________________________________

1 ten or tehen

2 ponwal

3 miuk

4 mnal

5 eip or ep

6 ten mesup or ta mesup

7 ponwal mesup

8 miuk mesup

9 mnal mesup

10 eip ne mesup or ep mesup

20 eipnal or epnal

_____________________________________________________________________

The other type of 5-cycle system included here, the (5, 20) type, is however common throughout New Guinea and parts of Melanesia and is found in both the NAN and AN language groups. The (5, 20) system is also referred to as the "finger and toe" or the "digit tally" system. Typically, this is such that we have four distinct monomorphemic numerals 1 to 4. The word for "five" is usually identical to, or cognate with, the word for "hand" or "arm". Occasionally "five" is a tally direction which contains a "hand" ("arm") morpheme and may have the meaning "hand to one side", or something similar. The second pentad, particularly the numbers from 6 to 9 (excluding the numeral 10 for the moment), usually has the construction represented symbolically as: 6 = 5+1, 7 = 5+2, 8 = 5+3, and 9 = 5+4; the word for "hand" usually appears explicitly in these compounds. Several variants of how these compounds are formed will be discussed below.

The word for "ten" normally contains a "hand" ("arm") morpheme and has the meaning "hands two", or something similar. "Ten" may be tally directions and we may have, thus, "hand side side" or "hands finished". Less often, "ten" is formed as an additive compound continuing the pattern for the numerals 6 to 9; thus 10 = 5+5, i.e "hand hand". In the true digit tally system, the numerals 11 to 19 contain a "foot" or "leg" morpheme indicating that tallying the toes is taking place. For example, 15 may be expressed as "hands two, foot one", or sometimes just "foot one" or "foot completed". On reaching 20 many systems have "one man" or "man completed", and so on, where it is understood that "man" means "the fingers and toes of one man". Indeed, some systems have this explicitly: "hands two, feet two" or "hands, feet completed", or occasionally "feet (legs) two" where "hands (arms) two" is implicit. The construction of higher numerals is either multiplicative, for example 40 = 2x20 and 100 = 5x20, or additive, for example 30 = 20 +10, i.e. "one man and two hands". Various examples of the digit tally system are discussed below together with several of their variants.

Two examples which are typical of the digit tally as described above are those of Sentani, located in Irian Jaya,[26] and Onjob, located in the Oro Province (PNG).[27] Both languages are NAN and the data are given in Table 14 below.

Table 14

Two Examples of Digit Tally Systems Belonging to NAN Languages

_____________________________________________________________________

Sentani Onjob

_______________________________________________________

1 mpai kema

2 be ameg

3 name towa

4 qeli rome

5 mehempai yanisiwi

6 mehine mpai yanisiforkema

7 mehine be yanisiforkema ameg

8 mehine name yanisiforkema towa

9 mehine qeli yanisiforkema rome

10 me be yani sisi

15 me be odo fe mpai amandi i ta kema ameg

20 u mpai yote kema

40 u be yote ameg

_____________________________________________________________________

For each of the systems above the numerals 5 and 10 contain a "hand" morpheme ("me", "yani" respectively) and the numeral 20 contains a "man" morpheme, "u" in Sentani and "yote" in Onjob. The second pentads of each system have the construction "5+n" where "n" takes the values 1 to 4. Onjob has a common variant where the numeral 6 does not explicitly contain the numeral 1 in its compound; in cases like this, "six" often can be translated as "one hand and another".

In Table 15 below, three examples, each AN, illustrate some of the variant ways in which the second pentad is constructed. The languages are Seimat,[28] located in the Manus Province (PNG), Gitua,[29] located in the Morobe Province (PNG), and Faga-Uvea,[30] a Polynesian Outlier located in New Caledonia. Each of the systems shown has four monomorphemic numerals 1 to 4, and "5" has a "hand" ("arm") morpheme "nim", "lima". The systems differ in the construction of the numerals 6 to 10 and we delineate each variant here. Seimat has, for 6 to 9, what I have termed a "5+n" construction. In this, "5" appears explicitly with each of the numerals 1 to 4 so that we have: 6 = 5+1, 7 = 5+2, 8 = 5+3, and 9 = 5+4; each of these compounds is formed without the use of copula or conjunctions. The numeral 10, in Seimat, has the construction "2x5" and does not continue the additive pattern of the numerals 6 to 9. In Gitua, "5" is "nimanda sirip" and has the meaning "hand half" or "hand side" and in the numerals 6 to 9 this appears explicitly. Each compound numeral has the form "5+c+n" where "5" is followed by a conjunction "c" (in this case "volo"), and then the corresponding numeral 1 to 4, thus 6 = 5+c+1, and so on. The numeral 10, in Gitua, does not continue this pattern and we have instead the construction "5x2". For Faga-Uvea, incidentally the only Polynesian language to have a 5-cycle counting system, we find that each member of the second pentad has the relatively unusual construction "n+c+x", in which the numeral 5 does not appear explicitly, and where the "n" takes the values 1 to 4, "c" is a conjunction, and "x" is a second pentad indicator. Thus 6, for example, is the compound "1+ona+tupu" where "ona" is a conjunction and "-tupu" is the second pentad indicator. The numeral 10, for Faga-Uvea, continues the pattern established for 6 to 9 and has the construction 10 = 5+c+x.

Table 15

Three AN Systems With Variant Second Pentads

_____________________________________________________________________

Seimat Gitua Faga-Uvea

________________________________________________________

1 tehu eze tahi

2 huohu rua lua

3 toluhu tolu tolu

4 hinalo pange fa

5 tepanim nimanda sirip lima

6 tepanim tehu nimanda sirip volo eze tahiatupu

7 tepanim huohu nimanda sirip volo rua luaonatupu

8 tepanim toluhu nimanda sirip volo tolu toluonatupu

9 tepanim hinalo nimanda sirip volo pange faonatupu

10 huopanim nimanda rua limaonatupu

_____________________________________________________________________

These three ways of constructing the second pentad do not exhaust the various possibilities; there is, nevertheless, across all the 5-cycle system types, a relatively small number of methods employed for compounding the numerals 6 to 9. One further and important method is illustrated below taking one NAN and one AN example each. These are, respectively, Panim and Bilbil, both located in the Madang Province (PNG).[31]

Table 16

Second Pentad Construction of the Form "x+n" in an AN and a NAN Numeral System

_____________________________________________________________________

Panim Bilbil

_______________________________________________________

1 olufan kete

2 elis oru

3 ized toli or tol

4 woalai pali or pal

5 mamagai nimanta

6 eben nahe koku kete

7 eben elis koku oru

8 eben ized koku toli

9 eben woalai koku pali

10 mamagunum nimanoru

_____________________________________________________________________

In each case the second pentad numerals 6 to 9 have the construction "x+n" where "x" is not identical to "5" and where "n" takes, respectively, the values 1 to 4. In the case of Bilbil, "x" is "koku(n)-"; whereas the "hand" morpheme "nim" appears in both 5 and 10, it does not appear explicitly in the intervening numerals. In the case of Panim, "x" is "eben" which is identical to the word for "hand"; this, however, does not appear in the word for 5 which is "mamagai". The latter, in fact, contains a "thumb" morpheme, "mamag-", which also appears in the word for 10.

Among the NAN languages of the New Guinea region, the (5, 20) "digit tally" system is the second most common type of counting system after the (2, 5)/(2, 5, 20) type discussed in Chapter 2. Of the latter, there are 109 altogether; by comparison 79 NAN languages have (5, 20) systems, just under one-fifth of the total sample. These are distributed among the various phylic groups as shown in Table 17.

Table 17

Distribution of (5, 20) Systems Among the NAN Language Phyla

_____________________________________________________________________

Small phyla 7

West Papuan 0

East Papuan 1

Torricelli 2

Sepik-Ramu 17

Trans-New Guinea 52

Total 79

_____________________________________________________________________

The distribution of the NAN (5, 20) systems throughout the New Guinea region is shown in Map 8 (Maps 8 to 11 are given at the end of this chapter). The majority are found in the coastal regions and in particular are located largely in the East Sepik, Madang, and Morobe Provinces of PNG.

Among the AN languages of New Guinea we have 39 classified as having a (5, 20) system. These are distributed as given in Table 18:

Table 18

Distribution of (5, 20) Systems Among the AN Language Groups

_____________________________________________________________________

Oceanic North New Guinea Cluster 19

Papuan Tip Cluster 11

Meso-Melanesian Cluster 1

Admiralties Cluster 1

Irian Jaya 2

Non-Oceanic Irian Jaya 5

Total 39

_____________________________________________________________________

The majority of these are confined to the North New Guinea and Papuan Tip Clusters and are thus located, as can be seen from Map 9, in the coastal and island regions of the Milne Bay, Morobe, Madang, and West New Britain Provinces. The non-Oceanic AN groups with (5, 20) systems are largely confined to the Maccluer Gulf region immediately to the south of the Vogelkop. In the remainder of Island Melanesia we find (5, 20) systems in both Vanuatu and New Caledonia. In Vanuatu, the 15 language groups which possess this type of system are mainly found in the islands of Ambrym, Epi, Erromanga, Tanna, and Aneityum (see Map 10). In New Caledonia at least 9 languages have (5, 20) systems and these are mainly found in the south of Grande Terre and in the Loyalty Islands. One of these, Faga-Uvea, is Polynesian (see Map 11 for these systems). Over the entire sample of 453 AN languages in this study we have, therefore, 63 which possess (5, 20) systems, or about 14%.

Among the 79 NAN systems and the 63 AN systems which possess this type of 5-cycle system there is, overall, a degree of uniformity in the etymological derivations of the words for 5 and 20. There are 51 NAN systems in which "five" contains a "hand" or "arm" morpheme; a further four systems contain a "thumb" morpheme and one has "palm". There are 26 NAN systems in which "twenty" contains a"man" morpheme; a further three have "hands two feet two". Among the AN systems at least 33 have a "hand" ("arm") morpheme in "five"; there is a further seven for which "five" does not contain a "hand" morpheme but "ten" does. There are 35 systems in which "twenty" contains a "man" morpheme and a further four which have "hands two feet two" or something similar.

The construction of the second pentad among all systems is restricted to a relatively small number of types and indeed the majority can be classified into just three.

Table 19

Showing the Distribution of Second Pentad Constructions of (5, 20) Systems Among the NAN and AN Languages

_____________________________________________________________________

5 + n 5 + c + n x + n

______________________________________________

Papuan (N = 79) 14 17 26

Austronesian (N = 63) 18 21 10

_____________________________________________________________________

3.4 SYSTEMS WITH A (5, 10) OR (5, 10, 100) CYCLIC PATTERN

The distinguishing features of the (5, 10) cycle system are that there are four monomorphemic numerals 1 to 4; 5 may be identical to, or cognate with, the word for "hand" or "arm"; the second pentad is formed additively although the word for "five" may or may not appear explicitly in the compounds for 6 to 9; there is a distinct numeral 10 and thereafter the system has a secondary 10-cycle and possibly a tertiary 100-cycle. Two examples, given in Table 20 below, one NAN and one AN, illustrate the nature of the (5, 10) system: these are Arandai, located in Irian Jaya,[32] and Kaliai, located in the West New Britain Province (PNG).[33] In both systems "5" appears explicitly in the construction of the numerals of the second pentad. For Arandai the construction takes the form "5+n" where "n" takes the values 1 to 4 and there is no conjunction. Kaliai, however, has the construction "5+c+n" for the second pentad where a conjunction is explicitly used. Both systems possess a distinct numeral 10 and the higher decades have an implied multiplicative construction, i.e. 20 = 2x10 or 10x2 but not 10+10.

Table 20

Examples of Numeral Systems With (5, 10) Cyclic Patterns

_____________________________________________________________________

Arandai Kaliai

______________________________________________________

1 onati ethe

2 ogi or ouge rua

3 aroi (ga) tolu

4 idati pange

5 radi lima

6 radi-onati or gendi lima ga ethe

7 radi-ogi lima ga rua

8 radi-aroi lima ga tolu

9 radi-tob lima ga pange

10 tobuti sangaulu

20 ogi-tobuti sangaulu rua

30 aroi-tobuti sangaulu tolu

_____________________________________________________________________

Two further examples, both AN, illustrate the (5, 10) system and, in particular, the common "x+n" construction for the numerals 6 to 9, where "x" is not identical to "5". The languages are Tolai, spoken in the East New Britain Province (PNG),[34] and Mota, located in the Banks Islands in northern Vanuatu.[35] These are given in Table 21:

Table 21

Two Examples of (5, 10) Systems With "x+n" Second Pentad Constructions

_____________________________________________________________________

Tolai Mota

_____________________________________________________

1 tikai tuwale

2 aurua ni rua

3 autul ni tol

4 aivat ni vat

5 ailima tavelima

6 laptikai lavea tea

7 lavurua lavea rua

8 lavutul lavea tol

9 lavuvat lavea vat

10 avinun sangavul

20 aura vinun sangavul rua

100 a mar melnol

1000 arip tar

_____________________________________________________________________

In both Tolai and Mota the numerals are compounded with a second pentad identifier, respectively "lap-/lav-" and "lavea", rather than the numeral 5 which does not appear explicitly. Both systems have distinct numerals 10, 100, and 1000 and indeed the cyclic patterns of each of these systems is (5, 10, 100, 1000). The Tolai numeral 5, "ailima" contains a "hand" morpheme "lima", however while Mota has "tave lima" for 5, this does not contain a "hand" morpheme which is "pane-". It is a distinctive feature of this type of system that "10" is normally a monomorphemic numeral unrelated to the words for "hand" or "arm".

The (5, 10) system is relatively uncommon among the NAN languages. The distribution between the various phylic groups is as shown in Table 22.

Table 22

Distribution of (5, 10) Systems Among the NAN Phylic Groups

_____________________________________________________________________

Small phyla 0

West Papuan 2

East Papuan 12

Torricelli 0

Sepik-Ramu 3

Trans-New Guinea 4

_____________________________________________________________________

We have, thus, a total of 21 such systems or less than 5% of our sample. Generally, the (5, 10) systems of the NAN languages of PNG are located in the coastal and island regions; five are located on Bougainville Island in the North Solomons Province and two are located in New Britain. In Irian Jaya, several are located in the Vogelkop region (see Map 8).

Among the AN languages the (5, 10) system is reasonably common. In PNG at least 45 language groups have it and it is distributed between the various clusters as shown in Table 23.

Table 23

Distribution of (5, 10) Systems Among the PNG Oceanic AN Clusters

_____________________________________________________________________

North New Guinea Cluster 20

Papuan Tip Cluster 7

Meso-Melanesian Cluster 17

Admiralties Cluster 1

Total 45

_____________________________________________________________________

These are located largely in the New Britain, New Ireland and Milne Bay Provinces (see Map 9). There is no evidence of this type of system occurring in the AN languages of Irian Jaya whether Oceanic or non-Oceanic. Nor is there evidence that this type of system occurs in New Caledonia. In Vanuatu, however, 68 of the 105 AN languages spoken there possess (5, 10) systems and these are located mainly in the Banks' Islands, Maewo, Pentecost, Efate, Malekula, and in most of Espiritu Santo (see Map 10).

As was the case with the (5, 20) systems, the (5, 10) systems are such that the construction of the second pentad numerals 6 to 9 fall largely into three main types as shown in Table 24.

Table 24

Distribution of Second Pentad Constructions for the (5, 10) Systems of NAN and AN Languages

_____________________________________________________________________

5 + n 5 + c + n x + n

______________________________________________

Papuan (N=21) 2 5 6

Austronesian (N=113) 10 19 74

_____________________________________________________________________

The large number (74) of AN languages which have an "x+n" construction for their second pentad is due mainly to the contribution of the Vanuatu systems, 64 of which have this type of construction.

Among the (5, 10) systems we find that a proportion are such that the word for "five" contains a "hand" (or "arm") morpheme. Of the 21 NAN languages 5 have this property, although one, Magi, has borrowed the numeral 5 from an AN source. Of the 113 AN languages, 40 are such that "five" contains a "hand" morpheme, that is about 35%. In some cases it is clear that for some languages the words for "five" and "hand" were, at some time during the past, identical but that over time a phonological shift has occurred. Several Milne Bay Province (PNG) languages are examples of this: Budibud has respectively "-lima" for "five" and "nima-" for "hand"; Kilivila has, repectively, "-lima" and "yama-"; Muyuw has respectively "-nim" and "nama-". In other languages there does not appear to be any relation between the two words.

3.5 SYSTEMS WITH A (5, 10, 20) CYCLIC PATTERN

The (5, 10, 20) system has features in common with both the (5, 20) and (5, 10) systems. There are four monomorphemic numerals 1 to 4 and the word for "5" normally contains a "hand" (or "arm") morpheme. The second pentad numerals 6 to 9 are additive compounds of the form "5+n" where "n" represents the numerals 1 to 4 respectively; there are variants of this as well. There is a distinct numeral 10 which does not usually contain a "hand" or "arm" morpheme. There is a distinct word for 20, usually containing a "man" morpheme, or something similar. Three AN systems are given below to illustrate this type of system: these are Manam, located on Manam Island in the Madang Province (PNG),[36] Mengen, located in the East New Britain Province (PNG),[37] and Pwapwâ, located on Grande Terre (New Caledonia).[38] The data are given in Table 25.

Table 25

Examples of (5, 10, 20) Numeral Systems for Three AN Languages

_____________________________________________________________________

Manam Mengen Pwapwâ

__________________________________________________

1 teke kena cac

2 rua lua celuk

3 toli mologi poecen

4 wati tugulu poeovec

5 lima lima nim

6 lima teke lima va kena ni cac

7 lima rua lima va lua ni celuk

8 lima toli lima va mologi ni cen

9 lima wati lima va tugulu ni ovec

10 kulemoa tangalelu paidu

20 tamoata giaukaena cac i kac

_____________________________________________________________________

Each of the systems above has a distinct numeral 10 and a distinct word for 20 meaning "man" or "being". The second pentad constructions differ however and represent the three main types of construction found: Manam has "5+n" for the compounds 6 to 9 while Mengen has "5+c+n" and Pwapwâ has "x+n".

Among the NAN languages this type of system is relatively rare with only 13 examples in our sample. These are distributed between the phylic groups as shown in Table 26.

Table 26

Distribution of (5, 10, 20) Systems Among the NAN Phylic Groups

_____________________________________________________________________

Small phyla 3

West Papuan 6

East Papuan 0

Torricelli 0

Sepik-Ramu 0

Trans-New Guinea 4

_____________________________________________________________________

There is no evidence that this type of system occurs in the East Papuan, Torricelli or Sepik-Ramu Phyla. The majority of the systems, 11 altogether, are located in Irian Jaya in the Vogelkop region, while two examples are found in the Madang Province (PNG) (see Map 8).

Among the AN languages we have a total of 46 which possess this type of system. Of these, 18 are located in PNG and are distributed among the clusters as shown in Table 27.

Table 27

Distribution of (5, 10, 20) Systems Among the PNG Oceanic AN Clusters

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North New Guinea Cluster 13

Papuan Tip Cluster 5

Admiralties Cluster 0

Meso-Melanesian Cluster 0

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These languages are largely confined to the New Britain, Morobe and Milne Bay Provinces (see Map 9). There is no evidence that this type of system occurs in either the Admiralties or Meso-Melanesian Clusters. In Irian Jaya, one Oceanic language has a (5, 10, 20) system while 8 non-Oceanic AN language groups do. In the remainder of Melanesia only in New Caledonia do we find this type of system with some 19 languages possessing it, the majority of these being located in the northern half of Grande Terre (see Map 11).

As indicated above, there is a limited number of variants in the way the second pentad of this type of system is constructed. As with all 5-cycle systems there is a small number in which the numerals from 6 to 9 are compounded using the construction "n+x" or "n+5" where "n" appears first in the compound and takes, respectively, the values 1 to 4 (thus 6 = 1 + 5, 7 = 2 + 5, and so on). However, generally speaking, the majority of the (5, 10, 20) systems have second pentad constructions which fall into just three main types: the "5+n", "5+c+n", and "x+n", as previously defined. Among the 59 languages having this type of system, the distribution of second pentad constructions is as given in Table 28. Out of all 59 systems there are 30 in which the word for "five" contains a "hand" (or "arm") morpheme.

Table 28

Distribution of Second Pentad Constructions of (5, 10, 20) Systems for NAN and AN Languages

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5 + n 5 + c + n x + n

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Papuan (N = 13) 4 2 6

Austronesian (N = 46) 12 22 7

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3.6 SUMMARY

Combining the various types of 5-cycle system we have a total of 113 NAN languages which possess one of the variants, that is just over a quarter of the total sample of 430. Of the AN languages, there are 222 which possess a 5-cycle variant, almost half the total sample of 453. The summary data for both the NAN and AN languages are presented in Table 29 below and it can be observed that the most common variant among the NAN group is the (5) or (5, 20) system, while among the AN group the most common variant is the (5, 10) system. It may be recalled from Chapter 2 that it is thought that the Proto Oceanic numerals formed a 10-cycle system and yet we find that almost half of the languages in the AN sample no longer have a 10-cycle system but have a 5-cycle system instead. Almost 36% of the sample do, however, retain a distinct numeral 10 and a 10-cycle although this is a secondary rather than a primary cycle.

Table 29

Summary Data for the 5-Cycle Systems: Their Distribution Among Both NAN and AN Language Groups

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(5)/(5, 20) (5, 10) (5, 10, 20)

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Papuan

Small Phyla 7 0 3

West Papuan 0 2 6

East Papuan 1 12 0

Torricelli 2 0 0

Sepik-Ramu 17 3 0

Trans-New Guinea 52 4 4

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Totals 79 21 13

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Austronesian

North New Guinea 19 20 13

Papuan Tip 11 7 5

Meso-Melanesian 1 17 0

Admiralties 1 1 0 Solomons 0 0 0

Vanuatu 15 68 0

New Caledonia 8 0 19

Polynesia 1 0 0

Irian Jaya 7 0 9

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Totals 63 113 46

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Map 8. Distribution of NAN 5-cycle systems (New Guinea)

Map 9. Distribution of AN 5-cycle systems (New Guinea)

Map 10. Distribution of 5-cycle systems (Vanuatu)

Map 11. Distribution of 5-cycle systems (new Caledonia)