natural use when rua tekau (two tens) is used for twenty, and toru tekau (three tens) for thirty, and so on. It prevents any misunderstanding as to how many tens are meant. But in former times, when tekau was used to denote twenty, it was not preceded by kotahi.

Another prefix to numerals is the word taki. This, as William's Dictionary states, “gives a distributive force to numerals” : hence takitahi = singly; takirua = by twos, two at a time; takitoru = by threes; and so on. Again, it is used before other words, “denoting that what is said applies to each one individually,” to quote the same authority: Ka takiomaoma ki te ngaherehere = Every one of them fled to the forest.

We will now give the modern system of Maori numeration, such as has been used since the early days of European settlement. The old system has been retained up to nine, but ngahuru, the ancient term for ten, has been rejected, and tekau substituted for it. This tekau is now the multiple of Maori numeration. Observe:—

And so on.

Here we note that ten has become the common multiple of Maori numeration. Apparently this change was made in order to assimilate the Maori system of numeration to that of the invading race. I can find no proof among the Tuhoe people that this system given above was used in pre-European days: hence it would appear that ten was not used as a multiple in former times.

It might be claimed that two was a multiple in the ancient

Maori system, but it resolves itself into a custom of counting by pairs, or braces; and it was not used in all cases—counting singly was also common. Persons were not counted in pairs, or braces, as was game, &c. These Natives would never have used ka wha pu to denote eight persons. But in one way a semi-binary system was used in counting persons; and this brings us to another prefix—viz., the word hoko. Hoko, as a prefix to numerals, is said by Williams to signify ten times the subjoined numeral; but when applied to persons the Tuhoe Tribe give it the value of twenty times the subjoined numeral, or ten times in pairs, whichever way you please to take it. Thus hokorua applied to persons signified forty; hokotoru = sixty; and so on. This system is similar to that of Aitutaki, described by Mr. J. T. Large at page 260, vol. xi, of the “Journal of the Polynesian Society.” He says, in the first place, that okotai takau (the aspirate is not used in the Cook Islands) stood for twenty, &c., and then states, “A correlative system of enumeration was also used indifferently with the above. This was distinguished by the prefix oko: for instance, okorua was twenty doubled, or forty; okotoru was sixty; and so forth, up to okoiva, which was 180; but it seems to have been confined to those limits.” This is exactly the Tuhoe case. From hokorua = forty, up to hokoiwa = 180, this system of counting obtained; but I have never heard hokotahi used to denote twenty, although it would seem that it was probably so used. Mr. Tregear looks upon hoko as a causative prefix, as hokowhitu = to make seventy.^*

It is probable that the prefix hoko was here used in both ways—viz., as signifying ten times the subjoined numeral, and also ten times doubled. Thus hokorua might mean either twenty or forty. In these cases the Maori could make his meaning clear by adding a word of explanation—either takitahi (singly) or topu (double—i.e., pairs). Thus hokorua takitahi would mean ten times two singly = twenty, and hokorua topu would be ten times two in pairs, or doubled = forty. ^† This point is not, however, yet quite clear. We have seen the value of the prefix hoko as given by Williams's Maori Dictionary—a most reliable work—but, still, my informants of the Tuhoe Tribe will not admit that hokorua signified twenty, and hokotoru thirty, and so on, but always double those figures, which would give the prefix hoko the power of multiplying the subjoined numeral twenty times, not ten times A confirmation of this comes from the east coast. The Rev. H. W. Williams, of Gisborne, informs me that he was told by

Mohi Turei, of the Ngati-Porou Tribe, that hokorua signified forty, and hokotoru = sixty, but that eighty was hokorua topu. This last term is very singular. Judging from the value of hoko before rua and toru, then eighty should be hokowha, as used by the Tuhoe people. It was surely a very strange break, or change, in this system of numeration to jump from hokotoru for sixty to hokorua topu for eighty. I cannot help thinking that this is an error. If, however; it was really the case that hokorua topu = eighty, then it is a proof that hoko really multiplied by twenty the subjoined numeral. Only one local authority has informed me that the hoko system was used in both ways, singly and doubly: as hokorua takitahi = twenty, and hokorua topu = forty, and so on. This would mean that Williams's Dictionary is correct that hoko multiplies by ten, and that hokorua topu simply means ten times two doubled, and not twenty times two. Anyhow, Vaux's statement was correct when he said that hoko was used for multiples of ten.^*

An examination of the Native methods of enumeration given in this sketch will show that several systems were employed—viz., counting singly, and the binary system of counting in pairs. There were also some differences in counting persons, and different words for various by-terms pertaining to enumeration: for example, the words kehe, taukehe, and tautahi all denoted an odd number. The terms paepae and tuma both mean an odd number in excess, as an incomplete ten or hundred. Tuhoe use the former word, and in this manner: kotahi rau, hokorua te paepae (one hundred, forty the excess) for 140. Tauhara and tauwhara are also terms for an odd number. By “an odd number” I do not necessarily mean the odd numbers three, five, seven, &c.; the terms are also used to denote (as in preserving birds) an incomplete ten. If eighty-three, or eighty-five, or eighty-six birds were put into a calabash, that vessel would be said to contain hokowha (eighty), ka whakarerea nga tauwhara (the odd ones are omitted).

The verb “to count” in Maori is tatau. Counting singly, as we do, would be described as tatau takitahi, and the dual method as tatau topu. Pu and topu bear much the same meaning—a pair, couple, brace. Takitahi, as we have seen, means—by ones, singly, once told.

It is possible that the last migration of Polynesians to New Zealand brought with them a somewhat different system of numeration to that in use among the original peoples, the descendants of Toi and the old-time tribes of these isles. They were certainly more advanced than the latter in some

arts—e.g., in cultivation. If so, this would explain some singular discrepancies and confusion noted when examining the methods of numeration employed by the Maori.

Before giving longer tables of the Maori systems of numeration, we offer a few remarks on some of the terms already quoted.

Rima = five. This term is said by many writers to be a survival of the primitive method of counting on the fingers. Ringa is the Maori word for hand, but linga and lima beai the same meaning in various Polynesian dialects: in Tahitian, rima = five, and also the hand; Hawaiian, lima = five, also the hand; Rarotongan, rima = five, also the hand; &c.^* The Maori still counts on his fingers in certain cases, as when repeating a genealogy, in order to count the number of generations from a certain ancestor.

Ngahuru.—This is the old Maori word for ten, now replaced by the term tekau. This word, recognisable under various letter-changes, is in use over a wide area in the Pacific: Rarotongan, ngauru = ten; Hawaiian, anaulu = ten days; Samoan, gafulu = ten. (See Tregear's Dictionary for many other comparatives.) Ngahuru is misspelt in Thomson's paper in the fifth volume of the “Transactions of the New Zealand Institute,” as also are many other Maori words. Ngahuru pu (= twenty) in Maori is literally ten pairs. Only once have I encountered this word in a different form among the Maori of New Zealand, and that was when an old man of the Tuhoe Tribe gave me the term tekau mahangahuru (or tekau maha ngahuru) as the ancient expression for thirty, in single counting. This is somewhat puzzling, and needs confirmation from other authorities.^† Tekau was twenty, and presumably the most likely term for thirty would be tekau ma ngahuru (twenty and ten), as there existed no special terms for thirty, fifty, seventy, and ninety in the Tuhoean system, according to my informants. But I have noted in various works that angahuru is supposed to have been an ancient term for ten in Polynesia (cf. Hawaiian anaulu above; though Tregear's example has anahulu). Hence I have thought that the expression above quoted should perhaps be written tekau ma hangahuru, which appears more natural when bearing in mind various Polynesian resemblances. Mr. J. T. Large states that the ancient term for ten at Aitutaki was ngaungauru. Hagavulu is used for ten in the New Hebrides.

An old-time Maori proverb is this: “Ngahuru kei runga, ngahuru kei raro”; which the late Sir George Grey translated

thus: “Never mind, I've ten teeth in my upper jaw and ten in my lower; hard or not, a hungry man can eat it.” It was used in reference to hard or tough foods.

For my own part, I but seldom theorize anent matters Maori. I am too busy at field work—i.e., collecting information from original sources. But I have a lone theory, and it concerns the word ngahuru, as used to denote ten. This I will proceed to give—hai kata ma te marea—though it leave me theoryless.

When a Maori proceeds to count on his fingers in the ancient manner he holds up his left hand open, fingers straight, in front of him. In beginning to count he takes hold of the top of the little finger of the left hand with the thumb and forefinger of the right. As he counts “one” he turns down the little finger until it touches, or nearly so, the palm of the hand. He then in like manner takes hold of the top of the next finger and turns that down as he counts “two,” and so on until he reaches “five,” when he turns the thumb in. Observe now the cream of my theory. All the fingers of the left hand have now become huru, or hurua—contracted, drawn in (from the verb huru = to contract, or draw in). This is one huru, or ringa huru; but it will not bear the plural of the definite article—i.e., nga—as a prefix. But he proceeds with his counting up to ten, which he tallies on the fingers of his right hand in the same manner as he did on the left, using the thumb and forefinger of the left hand to turn down the fingers of the right, but keeping the other three fingers of the left hand still closed on the palm. On completing the ten (ngahuru) he holds up both hands, with all fingers closed, as he repeats the word ngahuru. Here is where the plural comes in. Both hands (all the fingers thereof) are huru, or kurui—contracted; hence nga huru, or nga ringa huru—two contracted hands, ten fingers are huru'd. I want you to be careful of this theory, and treat it with all respect. I shall not make up any more: it is too exhausting.

John Fraser states in his excellent paper on Polynesian numerals that ngahuru and allied terms originally meant “the whole”—that is, the whole of both hands: hence ten. Judging from some of the terms quoted by Fraser, it would appear that huru alone meant ten: as in the Samoan e lua fulu (e rua huru) for twenty—literally, two huru; also in e fa ga fulu (e wha nga huru) for forty—literally, four huru. In the Polynesian isle of Bukabuka the term katoa, a word signifying “all,” is used to denote ten.

Another form of the word for ten is tingahuru. This form was used only when speaking of persons. A person asks, “Tokohia te whakareka?” (How many persons are there of

the invitation party?) and one might answer, “He tingahuru” (There are ten), or “He ti-ngahuru pea taua ope whakareka.” It is difficult to say what was the origin of this ti before the ordinary word for ten. Ti is a causative prefix in Maori, as in tiwaha, tirama, &c. Vaux states that the causative prefix whaka is placed before “ten” in order to form the ordinal, just as tua was used. He gives as examples tua-iwa and whakatekau (tekau for ten)—ninth and tenth. I have never heard whaka so used, but it may be employed thus by tribes with which I am not acquainted. (Yes; see Maunsell's Grammar.)

Tekau.—This term, as already observed, is now applied to ten, but the old men of the Tuhoe Tribe agree that in pre-European days it was applied to twenty only, never to ten. They also state that no decimal system, or multiples of ten, were in use among the Natives prior to the arrival of Europeans in these isles. Nor was any quinary system in use, although there was a vigesimal method of numeration, as we shall presently see.

In regard to the change made by Europeans in Native systems of numeration, we have on record cases of such made by early missionaries in Rarotonga, the Hawaiian isles, and elsewhere. Many writers have been misled by the modern system of counting among the Maori people of New Zealand, and have treated it in their essays as though it were the ancient system of the land, by which the value of their remarks or researches has been much impaired. I cannot prove that among all the Maori tribes of New Zealand tekau = twenty, but it was certainly so used among the Tuhoe, Ngati-Awa, and Ngati-Porou Tribes. In counting by pairs or braces the term ngahuru topu (ten pairs) was used for twenty.

We shall see that there are three main points to explain in Maori numeration, if not three systems—viz., counting singly, the binary, and the vigesimal or semi - vigesimal methods—not to speak of the modern system, or the changes made when speaking of persons. The binary or dual method was not used in counting persons, although the vigesimal system was, where hokorua = forty, and hokotoru = sixty.

As to tekau for twenty: In the far-distant Paumotu Group we find that twenty is rari takau—literally, “one takau,” rari meaning “one” or “alone.” In Tahitian, taau = twenty; Tongan, tekau = twenty, also fakakau = to put in scores or twenties; Marquesan, tekau — twenty; Mangaian gives takau = ten pairs; in Mangareva, takau = a double ten, takao = twenty. This is good evidence in favour of the Maori statement that tekau was originally used for twenty. Mr. J. T. Large states that at Aitutaki Island twenty was expressed by the term okotai takau (okotai is hokotahi in the New Zealand dialect).