Volume 39, 1906
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### The Vigesimal System of Enumeration (Table No. 1).

It appears to me that at some period of their history the Maori must have used a vigesimal numerical method—a system of counting by scores, or twenties. I shall include in this paper a table showing the method so far as I have been able to ascertain it from my local Native friends. It will be observed that there was a special term (tekau) for twenty, but none for thirty; a special term (hokorua) for forty, but none for fifty; a special term (hokotoru) for sixty, but none for seventy; and so on. Thirty was twenty and ten; thirty-one was twenty, ten, and one; and so on to thirty-nine. Forty was again a special term, then forty and one, then forty and two, and so on to forty-nine. Fifty was forty and ten; fifty-one was forty and ten and one, &c. (See Table No. 1.)

In the “Journal of the Polynesian Society,” vol. x, p. 101, Professor Cyrus Thomas gives a short paper on the vigesimal system of enumeration. In it he observes that traces of a

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former system of vigesimal numeration have been observed in Oriental lands—in south - eastern Asia, Cambodia, and Malaysia—but long overlaid by the decimal system. The Cambodia system, as given by Aymonier, much resembles that of the Maori. There are characters for each of the nine digits, for twenty, and for a hundred (and presumably for ten). “The character for twenty is distinct, and not two tens. In order to indicate thirty-seven, there is first the character for twenty, then for ten, and last for seven.” All this is the same as the Maori system. He goes on to say that forty is two twenties, sixty is three twenties, &c., there being a separate character for a hundred. “A mingling of the two systems is apparent in some of the examples given by Aymonier,” &c. Now, this is just the Maori case. We note how the vigesimal and the topu or binary methods are sometimes confused by Natives, not in the special terms for scores so much as in the intermediate items. Doubtless, however, much of this confusion arises from the fact that these old-time methods of the Maori have been laid aside in favour of the decimal system introduced by Europeans, albeit the latter system is expressed in purely Maori terms. The older generation of living Natives can only recall the old-time numerical terms by an effort of memory; indeed, some have forgotten many of them. The younger generation know practically nothing of these matters. It is when an old Native is repeating tribal traditions, &c., that one hears quotations from the old numerical systems, but seldom under other circumstances.

Professor Thomas states that among the Maya people “The numbers from one to eleven had specific names, but from twelve to nineteen [were formed] by the addition of units. There was a specific name for twenty, four hundred, and four thousand. Numbers from twenty to four hundred were formed mostly by twenty as the multiple, and units.” He notes some confusion, however, and evidence of the quinary and decimal systems.

In vol. xi. of the “Polynesian Journal” Mr. Large mentions a vigesimal system used by the Natives of Aitutaki, which is practically the same as that of the New Zealand Maori, the same terms being used, although the Cook Islands dialect has lost the h and aspirated w, the w having become v. In that system okorua = forty, okotoru = sixty, and so on.

In the same volume Mr. Percy Smith states that the-Natives of Niue counted fish by twenties, te kau (or “two tens”) being the term used, though it would appear that tekau was a specific term for twenty, and not “two tens.” Was kau = ten, and te a plural, or “two,” that tekau should be given as = two tens? Elsewhere it appears to be a special term for twenty. Tekau does not appear in the Niue

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vocabulary published by the Polynesian Society, neither does te; kau is given as meaning “company, troop.” Kau is probably the root of the word, te being the definite article.

Since writing the above I have unearthed a long-buried copy of Maunsell's Maori Grammar, 3rd edition, 1882. I quote a few remarks therefrom to show how numeration differed in some districts, the result, perhaps, of tribal isolation. He explains the modern method of numeration by multiples of ten, but states in a note that “It should be here noticed that this is the new mode of reckoning brought in by Europeans, and now fast spreading over the land. The old mode is not so convenient, but it is often heard; 240 would, according to it, be thus expressed: kotahi rau ma rua— literally, one hundred and two. Rua here stands for (twice ten) twenty doubled. 250 would run thus: kotahi rau, ma rua pu, tautahi—one hundred and two double, and a tautahi (odd one).” Now, the above was the method of counting in the Waikato district. Observe that it was the dual system, couples or pairs being always implied, while ma rua, “and two,” is made to serve for “and twenty” (couples understood). In the second illustration he gives the term pu, signifying “pairs,” or “twice told,” or “doubled”—thus, one hundred, and two pairs (for twenty doubled); this “two pairs,” or “two doubled,” being perhaps an abbreviation, though noticeable all over the Island. The term tautahi is used in a similar manner among Tuhoe. It is usually applied to a single odd number—e.g., e waru pu, tautahi, for eight brace (or couples) and an odd one = seventeen. In Maunsell's example, however, it stands for ten, or an odd ten.

Maunsell, in giving the modern system, stated that hokorua is used for twenty, but explains in a note, “The Maori mode of counting has always heretofore been by pairs: thus hokorua, twenty, stands for twenty pair—i.e., forty—and so on. When they wish it to be understood singly they postfix takitaki to the numeral adjective—i.e., hokorua takitaki =twenty.” This takitaki may be a misprint for takitahi, the term in general use to denote “singly,” or “by ones.” Here we have evidence that hokorua, hokotoru, &c., really mean twenty, thirty, &c., but that the term topu or pu (pairs) is understood; unless the expression takitahi be added, in which case hearers understand that counting singly is meant. Thus evidence accumulates to show that Maori numeration was dual in its character, and that the term topu (or pu) was by no means always used when employing that system, but was understood to be implied; also, that it was necessary to use the term takitahi to show that single numeration was meant.

Maunsell goes on to say that among the Ngapuhi Tribe

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ngahuru was used for ten and tekau for eleven, while in the centre of the Island ngahuru and tekau both represent ten. Judging from Polynesian comparatives, the use of tekau, both for ten and eleven, seems to be of local origin, though it may, strictly speaking, have meant ten pairs He also gives a variation in the form of the distributive prefix, or a plural form thereof, where tataki is used for taki, the example being, Kia tataki rua pu nga utu i te tangata (Let each man have four payments). In speaking of the ordinals he gives three ways of expressing such—(1) By tua prefixed to the cardinal, as tua toru = third; (2) by whaka prefixed, as whakatekau = tenth; (3) by the simple cardinal with the definite article, as te wha=the fourth. The first and third of these modes have been given as in use among the Tuhoe Tribe, but whaka I have not heard so used. Was it used before ten only, or might it be used before any of the digits?

Having now (7th January, 1906) obtained some further information anent Maori numeration, I proceed to add the same to above notes.

Tekau.—Several old Natives of the Tuhoe and Ngati-Awa Tribes confirm the statement that tekau was formerly used to denote twenty, and was not used for ten. As kau seems to have been a Polynesian word meaning “collection” or “assembly,” then the expression would probably have been originally te kau = the whole, or the assembling of the ten fingers and ten toes, te being the definite article singular.

Table No. 1: This shows the ordinary mode of counting singly, as formerly used. It includes, in a singular way. a vigesimal system—that is to say, it is partially vigesimal. It has the special term for ten which, however, was not used as a multiple. It has a special term for twenty, but none for thirty; a special term for forty, but none for fifty; for sixty, but none for seventy; and so on. Thirty was “twenty and ten.” Thirty-one was “twenty ten, and one”; and so on to thirty-nine. Forty was a distinct term (hokorua), and then another twenty was commenced It will be observed that the vigesimal system was never carried beyond 180 (hokoiwa), or nine twenties, except in conjunction with the rau (hundred): e.g., kotahi rau, hokowhitu, for 340—i.e., one hundred twice told and seventy couples. A common form, however, was to abbreviate such terms: as kotahi rau ma whitu, or he rau ma whitu, for 340; kotahi rau ma rua, for 120 doubled; and so on.

I have consulted a great many Natives as to the value of the prefix hoko, and the majority state that this prefix conveyed the meaning of twenty times the subjoined numeral. Some, however, maintain that it merely implied ten times the

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subjoined numeral when the word takitahi (single, or singly) was added: as hokorua takitahi=twenty; hokotoru takitahi=thirty; and so on.

Confirmation has been received regarding the term tekau maha ngahuru for thirty, not only local, but also from the Ngati-Awa Tribe. I am unable to account for the syllable ha.

Table No. 2: This gives the dual system of numeration formerly in use among the Maori people, the counting by pairs, which was a common custom. This form of binary numeration was used only for game, baskets of food, and so forth, but was not applied to the genus-homo. I have remarked above that the binary method was not used when counting persons. This remark needs some explanation. Certainly the dual method of numeration, as given in Table No. 2, was not used for persons, but another form of counting in pairs* was used for persons. Apart from the matter as to whether the prefix hoko multiplied the numeral ten or twenty times, there were other expressions used which doubled the number given. We have seen that kotahi rau ma whitu (one hundred and seventy) was used for 340. This was certainly applied to persons, as in giving the numbers of a war-party. This, and hokowhitu for 140 men, were such common terms that the word topu does not seem to have been employed to denote the fact that the number given meant so-many couples. But with other numbers the evidence seems to be in favour of the term topu, or takitahi, having been used: as kotahi rau takitahi (one hundred, singly, or once told), and E wha rau topu taua ope (That party consisted of four hundred [persons] twice told, or doubled). These expressions are used when speaking of persons, and seem to have been so used formerly. Many of my old Native friends say, “Kaore i takirua te tatau mo te tangata”—i.e., persons were not counted in pairs. I believe they mean that, when actually counting a number of persons, the system given in Table No. 2 was not used. And it certainly was not. A person would not have counted persons in this manner—ka tahi pu (two), ka rua pu (four), ka toru pu (six), &c.—as he would in counting game, &c.; nor would he have said ka toru pu, tautahi, for seven persons. He would count them singly, and for seven persons he would have used tokowhitu (see Table No. 3). But if he had counted, say, 240 persons, and was asked how many there were, he would have replied, “Kotahi rau, hokorua”—one hundred (topu understood) and a hokorua; or “Ko tahi rau ma wha”—one hundred and forty (topu understood). In

[Footnote] * Or not exactly in pairs, as in Table No. 2, but the doubling of stated numbers, a “twice-told” mode.

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stating numbers between one hundred and another, the terms ma rua, ma toru, ma wha, &c, were used to imply “and twenty,” “and thirty,” “and forty,” and so on, though, strictly speaking, the expressions mean “and two,” “and three,” &c. Another ancient method of stating 240 was kotahi rau, hokorua te paepae—one hundred (topu understood), the excess a hokorua.

But to return to Table No. 2: In this is shown the method employed in counting by pairs—i.e., the terms used for every number from 2 to 102, and a few of the leading figures from that number up to 1,000, which the reader will have no difficulty in following. But it must be mentioned here that a person engaged in counting a number of articles by the dual method would not make use of all these terms; he would not count the odd numbers, where the expression tautahi is employed. The terms for odd numbers are merely inserted to show what words express such numbers. Such are used only to express the total when that total contains an odd number. Observe: A fowler visits his bird-snares every morning in order to collect the birds. Having completed his round, he proceeds to count the birds taken. This he does by taking up two birds at a time and laying them aside. For the first brace he counts “Ka tahi pu”; for the second, “Ka rua pu”; for the third pair, “Ka toru pu”; for the fifth, “Ka rima pu”; for the tenth brace, “Ngahuru pu.” Here the word ka is dropped, but it is sometimes resumed for eleven brace, as ngahuru-pu, ka tahi pu, and so on; ngahuru pu, ka iwa pu, for thirty-eight. It is not usual to use ka before hokorua, hokotoru, &c., but it is sometimes resumed after them: as hokorua, ka tahi pu, for forty-two; hokorima, ka whitu pu, for 114, &c. Indeed, I am inclined to believe that this was the more correct way of expressing numbers when actually engaged in counting.

Suppose our friend the fowler has taken seventy-seven birds: he goes on counting by the brace (pu) up to seventy-six—hokotoru, e waru pu (or ka want pu); then, casting the remaining bird on the heap, he says, “tautahi” (an old one). The number of birds taken is expressed by hokotoru, e waru pu, tautahi (sixty, eight brace, odd one).—Q.E.D.

The singular feature of this system of counting is the combination of the dual and vigesimal systems. It is purely dual up to twenty, but from the number twenty-two onwards to thirty-nine the numbers hinge upon twenty—as “ten brace, one brace,” “ten brace, two brace,” and so on—until the next twenty (i.e., forty) is reached, where we note the special term hokorua, which again has pair after pair added to it until sixty (another special term) is attained, and so on to 199.

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For 200 we find a new term employed—viz., the rau, or hundred, doubled. Upon this new base the pu and hoko systems are built up until 400 is attained, when we find the term rua rau (two hundred—brace understood) employed. The same system is repeated until 600 (toru rau), and so on. In actual counting, a person would probably say “Ka rua rau, ka toru rau,” &c., and not “E rua rau, e toru rau.” But if simply stating a number, not counting, he would probably use the particle e: “E hia nga kete riwai i a koe?” (How many baskets of potatoes have you?) and the answer would be, “E rua rau,” or “Hoko toru,” or whatever the number might be. But if the questioner used the verbal particle ka, then the answer would be preceded by that term: “Ka hia au manu?” Answer, “Ka rua rau,” or “Ka wha pu,” &c.

We have seen that paepae was used to imply an excess number. For instance, in counting by the dual system, 460 would often be given as e rua rau, hokotoru te paepae (two hundred—pu or topu understood), the paepae being sixty. This term for an excess number seems to have been used between hundreds only—i.e., for numbers between 100 and 200, between 200 and 300, and so on. It is said to have been used in counting objects (game, &c.) only, and not in counting persons. The word paepae means the odd or excess numbers stretching forward towards the next hundred. Pae means ‘a step; direction; perch; to lie across; lie ready for use,” &c. Whakapae = to lay across. Paepae and paewai = threshold. Paepae is the step towards the next hundred. Of these numbers between hundreds an old Native remarked, “E pae tonu ana, kia tae ki te rau, kua kore e kiia he pae” (They are all in the pae stage; when the next hundred is attained, the term pae is not applied). But it is again employed when the next hundred is commenced.

Another common expression for an excess number is tuma. Thus ngahuru tuma means “ten and an excess”; and it may be used for any number from eleven to nineteen inclusive. Kotahi rau tuma stands for one hundred and an excess, and may be used for any number from 101 to 199 inclusive. Such usage is equivalent to our expressions “twenty odd” and “one hundred odd,” &c. These illustrations are from the single method of counting. Maunsell gives an illustration from the dual method—viz., e rua mano ma wha, hokorima te tuma, for 4,900; but literally it is “two thousand and four, hokorima”—the words “hundred doubled,” or “hundred pairs,” are omitted after the word “four,” but are understood. In the modern system of counting by multiplying by ten we often hear the word tuma for, the excess numbers between tens. But then, ten was not a multiple in the ancient system, nor was tekau used for ten. As old Tutaka expressed

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it, “Tekau as a term for ten is a modern usage. It was the white man and his books that made it known to us.”

Makere is another term used to imply an excess number. Ngahuru makere (ten odd) seems to bear the meaning of “ten onwards,” and may be used in relation to any number from eleven to nineteen inclusive. When explaining to me the meaning of the expression ngahuru tuma, a Native said, “Mo te tekau makere tena karanga” (That term is used to denote ten onwards). Here, of course, tekau is used with its modern meaning of ten.

And again, we have the word rerenga used in a like manner. Rerenga is a verbal noun (rere = to run, flow, &c). Kotahi tekau, e whitu te rerenga, means “one ten, the balance or excess being seven.” This, again, is the modern tekau = ten. Again, kotahi rau me nga rerenga stands for “one hundred and the balance” (or excess over 100).

The terms tauwhara and tauhara bear a similar meaning of excess numbers. When explaining to me the ancient binary system of counting, a Native said, “Game was so counted in former times, when the birds or rats were taken from the snares, but when they were potted in calabashes the odd numbers were omitted, and eighty-five birds would be styled a hokowha (eighty). (Kia maoa rawa nga manu, kia uru ki te ngutu iti, ka whakarerea nga tauwhara.)

Still again, we have the terms kehe and taukehe as meaning odd numbers. Taukehe is sometimes used in place of taukehe, when counting by the dual method: hence ka rua pu, taukehe, would be used for five; ngahuru pu, taukehe = twenty-one; and so on. Kehe is often used to express an odd number. When looking at a hut in course of erection a Native said to me, “E he ana nga heke, kua kehe” (The rafters are wrong, there is an odd one). It is a Native custom to always put an even number of rafters on either side of a roof. It is a sign of bad luck to put an odd number.

Williams's Maori Dictionary gives the following words not used among the Tuhoe Tribe: Hara = excess above a round number—kotahi rau, e iwa nga hara. Hemihemi = excess over a definite number—kotahi rau ma whitu, hemihemi (one hundred and seven and over).

Whakamoe (or whakamoe mătăa) is an expression employed to denote the counting of game in braces, laying aside each brace as counted—Kai, te whakamoe a Turei i nga manu o te taha.

In counting koko (tui) birds prepared for preserving, a pu or brace consisted of four birds (I am not a Milesian)—i.e., they were set aside by fours, but the four were only called one pu. Possibly this was on account of the smallness of

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the bird. (Mo te koko, kia makiritia, ka penei te karanga, he pu koko, mo te wha takitahi tena karanga).

Although I have given in Table No. 2 that form of dual numeration which I believe was generally used in this district, yet some of my authorities differed from it in their accounts; for instance, one man gave a different method of counting from twenty-two to thirty-eight inclusive. Observe:—

 20. Ngahuru pu. 22. Ngahuru ma tahi ten and one; “pu” understood). 24." rua " two; “brace” understood). 26." toru " three "). 25." wha " four "). 30." rima " five "). 32." onó " six "). 34." whitu " seven "). 36." waru " eight "). 38." iwa " nine "). 40.Hokorua.

Here we see pairs added to ten pairs until thirty-eight was reached, after which it was the same as Table No. 2—pairs added to twenties.

Another differed in his term for twenty only. He used tekau instead of ngahuru pu, and added pairs to it as in Table No. 2—tekau, kotahi pu (twenty-two), tekau, e rua pu (twenty-four), &c. This man is one of those who maintain that persons were always counted singly. He says that when speaking of persons, hoko multiplies the subjoined numeral by ten only—hokorua = twenty, hokotoru = thirty, &c.—while it multiplies the numeral by twenty in counting game, &c. Another old Native, a man of much knowledge, also maintains that hoko was used in both ways—i.e., takitahi and takirua—which supports Williams's Dictionary. Thus hokorua takitahi would be twenty, while hokorua topu would stand for forty. This latter (Native) authority states that if in stating a number a person simply said “Hokowha,” he would be asked, “Hokowha aha?” (Forty what?) and the reply would be “Hokowha takitahi,” or “Hokowha topu,” as the case might be. Also that, in using the dual method, the terms hokorua, hokotoru, &c., should really be followed by pu, but that it is usually omitted. Again, he states that prefixes were often omitted—e.g., ngahuru pu, tahi pu (twenty-two); ngahuru pu, rua pu (twenty-four), &c.—when counting.

When, in counting his bag, a fowler found he had taken, say, 105 or 107 birds, he would often wait until he had made the number up to, say, 110 before returning home, to abolish the taukehe.

One of my old Native authorities is confident that in former times the Tuhoe people counted up to 1,000 readily,

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but did not carry the system beyond that number. Still, it is clear that they could express any number up to 2,000 by the use of the term topu—as E whitu rau ma rima topu = seven hundred and fifty pairs = 1,500; kotahi mano topu = one thousand pairs = 2,000. Higher numbers were, he says, expressed by such terms as tuni, &c. (see ante).

The words nui and maha mean “many,” the latter being prefixed by toko when applied to persons. Iti is sometimes used as = “few,” though, strictly speaking, its meaning is “small” (tokoiti) when applied to persons. Tokohinu = some. Ouou, ruarua, and torutoru are also used for “few,” and are sometimes, not always, given the prefix tokoi.e., when speaking of persons. The two last expressions are, it may be observed, formed by doubling the words rua (two) and toru (three). In Williams's Maori Dictionary we find rūrŭa = both equally. (E tika rurua ana raua=They are both equally correct.)

We have seen that măno is used for 1,000—a specific term for that number; also that it is employed with a more vague sense—“numberless,” or “multitudinous”—though often coupled with other expressions, as mano tini or mano raua ko tini, mano tini whaioio. The last is sometimes merely tini whaioio. Williams gives hea=multitude, majority; and mano tuauriuri = very many. The numeration terms ngera, makehua, maioio, rea, given by Maunsell, I have not heard used, nor yet the expression tini whakarere. They are probably peculiar to tribes of the Waikato, or northern districts. Tini makehua is a peculiar term. I cannot refrain from thinking makehua allied to makahua, a generic term for stones. Tylor, in his “Anthropology,” gives some account of the origin of numeration and ciphering, showing how many people reckoned with stones used as counters; as also the origin of the Latin calculare, and our word calculate, from calculus = a pebble.

As observed, none of the Maori terms for the digits seem to have any connection with the names of the fingers, although the word for five (rima) is apparently an old-time Polynesian word for hand. The names of the fingers are takonui (thumb), takoroa (forefinger), manawa, mapere, toiti. These are termed the tokorima a Maui (the five of Maui). The prefix toko is employed because the five were persons—i.e., the personifications of fire. For these were the Fire Children of Mahuika, the Maori fire-goddess, who were destroyed by Maui when he obtained fire for man. If, when offering food to a Native, you apologize for the lack of knife and fork, he will say, “Never mind, I have the tokorima a Maui.”*

[Footnote] * Sometimes simply tokorima. Ex., “E aurakina nei e oku tokorima.”

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It is a very ancient myth, and curious withal, that of the origin of fire—that is to say, of the Fire Children. “It was in an age long past away, before man was, that the thought came to the son of Tangotango [i.e., to the sun] that he would send his child to the lower world to convey to his descendants there a great boon, the blessing of fire. Hence he said to his child, to Auahi-tu-roa, ‘Go you to convey a boon to our descendants in the world.’ And his son asked, ‘How shall I give it?’ The sun replied, ‘Give them fire’ [tokorima]. So Auahi-tu-roa descended to the earth. He came to Mahuika, younger sister to Hine-nui-te-Po [goddess of Hades], and ere long she gave birth to five children, whose names were Takonui, Takoroa, Manawa, Mapere, and Toiti. Those children were the Fire Children.” Here we see how fire originally came from the sun. But this is digression. Return we to our numeration.

Williams gives a word I have not before met with, makiu = very numerous. Ex., “Tuauriuri whaioio, makiu, makiu.”

The modern method of adding units to 100 is by means of the conjunction ma: kotahi rau ma whitu—one hundred and seven. But in the old system of single counting this phrase seems to have stood for 170. Kotahi rau ma rua = one hundred and twenty (literally, one hundred and two). This method was used between hundreds, to express the odd tens, and is the only item in Tuhoe numeration of pre-European days that is decimal in its nature.

An ancient way of adding units to hundreds was by using the particle e—kotahi rau, e whitu, for 107. Kotahi ma whitu was not only an exact term for 170, but also a vague expression employed for any number of persons between 100 and 200. Rau ma whitu was used in the same manner, applied to a war-party, or company of travellers. It simply meant between 100 and 200. In like manner the term hokowhitu was used in the same vague way, when applied to a company of people. As my informant put it, “Kia eke rawa ki te rau, katahi ka karangatia.” It was only when the hundred was attained that the term was altered.

In that very singular work, “Te Ika a Maui,” by the Rev. R. Taylor, are some curious remarks anent Maori counting. He says, “The old Maori way of counting was evidently at first by the fingers up to ten, then a shake of both hands was given, which signified one ten—this was called a nga huru, or the entire ten fingers; one hand being shaken implied five, or the half; ten shakes of the two were 100; and so on. Thus kotahi was one finger; ka rua, two, &c.; ka tekau, ten: then a shake of the hands was given

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—this was nga huru, the whole ten fingers; tekau ma tahi, eleven; tekau ma rua, twelve, &c.; rua tekau, twenty; and so on to kotahi rau, 100, and kotahi mano, 1,000: beyond that all numbers were mano tinitini … Unless they added takitahi (once told) to these shakes they meant double: thus kotahi rau would signify 200, unless they said kotahi rau takitahi (a hundred once told). Pu also signifies counting by pairs, unless qualified by the word topu: thus kotahi pu topu is simply two, but e rua pu is two pairs, or four; ngahuru (ten) is thus twenty. The word hoko signifies the doubling of twenty: nga hoko rua, forty; nga hoko toru, sixty; nga hoko tekau, two hundred … kotahi rau hoko whitu, one hundred doubled and seven twenties. Topu also signifies a pair doubled, or four.”

The items in the above which arrest our attention are: The shaking of the hand, or hands, to show that five, or ten, is complete; and the use of tekau for ten, which the reverend author seems to imply was an ancient custom. It would be interesting to know what tribe these notes were obtained from. His remarks on the terms pu and topu are peculiar. I cannot see how topu qualifies pu. Pu means a pair, and requires no qualification. Kotahi pu topu sounds very tautological, while hoko scarcely signifies the doubling of twenty, but the multiplying by that number.

We may note that the Maori had no knowledge of ciphering, or any form of abacus, so far as we know. They possessed, apparently, one only mnemonic aid to memory in their genealogical staves. This was a piece of hardwood about 1 in. in diameter and 3 ft. or so in length. It had on one side a series of square-edged notches cut in it, the pieces of wood left between the notches being about ½ in. or ¾ in. These represented each a generation. These staves were, if necessary, used in a boustrophedon manner. They were but a crude aid to memory, and their use does not appear to have been very common.

The word mutu is sometimes used after a round number to show that no excess exists—e.g., kotahi rau mutu (one hundred and no excess).

Ngahoro is another term used to imply an excess number: thus hokorua ngahoro is equivalent to hokorua makere, &c., and means a hokorua and an excess number. If hokorua is used in the topu sense, then the above expression may be used for any number from forty-one to fifty-nine inclusive.

Table No. 3: In this table we see the old-time Maori method of enumerating persons. The prefix toko, used only when speaking of persons, has already been noted, as also the terms tingahuru and tekau.

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Although the topu or binary system of counting, as given in Table No. 2, was not applied to persons, yet the double hoko and rau methods were. Thus hokorua was usually employed to denote forty persons, hokotoru for sixty, and on to hokoiwa for 180. The single hoko system appears to have been sometimes employed when stating an excess over a round number (see Table No. 3 for examples). The single rau, or hundred, seems to have been seldom applied to persons, 100 being expressed by hokorima. But 200 was kotahi rau topu—i.e., one hundred doubled, or one hundred pairs. The word topu was not actually used, as a rule, but was left to be inferred. Thus 140 was termed a hokowhitu; 340 was a rau hokowhitu.

I have given many terms in Table No. 3, so that the reader may know the exact terms for precise numbers; but it must be here explained that the Maori usually gave round numbers for persons, and seldom expressed exact numbers between twenties. He would use the term hokorua makere for any number between forty and sixty, and would not specify the excess unless under peculiar conditions—e.g., in answer to a question. In like manner the expression kotahi rau tuma might be used for any number between 200 and 300, unless he employed the takitahi method, in which case it might stand for any number between 100 and 200. The table shows the forms employed to express the excess numbers, when required. The number 101 might also be expressed by hokorima, kotahi te tuma; 102 by hokorima, tokorua te tuma; and, so on to 119 = hokorima ngahuru ma tahi te tuma (or paepae); 122 might be given as hokoono, tokorua te tuma, and so on; 130 as hokoono, he ti ngahuru te tuma; 131, hokoono, ngahuru ma tahi te tuma, and so on. The different terms used to denote a given number are perplexing in the extreme. Possibly different methods, or different expressions, were formerly used among different tribes, and the modern Maori has confused them.

If the single hoko and rau terms are used, then the system of numeration may be termed decimal; but if the double hoko method be employed (as hokorua = forty, &c., and kotani rau = 200), then the system is vigesimal.

Regarding the numbers eleven to nineteen in Table No. 3, these were given to me by Te Puia, of Tuhoe, as—eleven, ti ngahuru, kotahi; twelve, ti ngahuru, tokorua; thirteen, ti ngahuru, tokotoru; nineteen, ti ngahuru, tokoiwa. He also gave — fifty-one as hokorua, ngahuru takitahi, kotahi; fifty-two as hokorua, ngahuru takitahi, tokorua; fifty-three as hokotorua, ngahuru takitahi, tokotoru; and so on to hokotoru = sixty.

The Ngati-Awa Tribe, according to Matutaera Hatua,

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objected to applying the topu system of counting to persons because it interfered with their kawa tapu. Probably this was on account of food-supplies being counted by the topu or binary system. As almost all local authorities agree that persons were counted singly, and not by the tatau topu, I feel certain that the double hoko and rau methods (hokorua = forty, &c.; kotahi rau = 200) are not viewed by Natives as being binary in their nature, but that that expression is applied only to the system of counting in pairs, as given in Table No. 2; and, moreover, that the vigesimal (or double hoko) method was a system of numeration known to, and used by, the Polynesian peoples in times long past away. Thus, the reader will bear in mind that the vigesimal system, as given in Table No. 3, seems to have been that in common use for enumerating persons. The variant forms, as kotahi rau takitahi for 100, kotahi rau ma rua and kotahi rau, hokorua takitahi te paepae, for 120, &c., do not seem to have been so much used, at least according to evidence now obtainable.

In regard to Table No. 1, it seems probable from the evidence of many Natives that the terms hokorua makere, hokotoru makere, and so on, were also much more commonly used than a perusal of that table would lead one to suppose. I have given the definite terms between twenties, in order to place my notes on record, and to render the table complete; but a Native would very often say hokorua makere for any number from forty-one to fifty-nine inclusive. This is equivalent to our system of counting by scores, when we say, “Three score odd,” &c.

A singular form obtains among some of the Waikato Natives, of expressing the numbers twelve to nineteen in the numeration of persons, as—tekau ma tokorua, for twelve; tekau ma tokotoru, for thirteen; and tekau ma tokoiwa, for nineteen.

Puihi Maru-tawhao, an old man of Tuhoe, has given me the following notes lately: The pu koko (see ante) consisted, in this district, of six birds. These were, presumably, considered equal to two pigeons or kaka. Both the single and double hoko systems were formerly used here—e.g., hokorua takitahi stood for twenty, and hokorua topu for forty. He says that hokotahi was not here used. Fish, as well as birds, were counted in pairs—i.e., the binary method was employed; but baskets of kumara, &c., were counted singly.

And here endeth such notes as we have collected anent the systems of enumeration employed by the Tuhoe Tribe in days of old. They are not remarkable for clearness, but represent, nevertheless, much work in collection, and close questioning of many persons.