As if to emphasize the rarity of this method of forming numerals, the Jiviros afterward discarded the last five of the above scale, replacing them by words borrowed from the Quichuas, or ancient Peruvians. The same process may have been followed by other tribes, and in this way numerals which were originally digital may have disappeared. But we have no evidence that this has ever happened in any extensive manner. We are, rather, impelled to accept the occasional numerals of this class as exceptions to the general rule, until we have at our disposal further evidence of an exact and critical nature, which would cause us to modify this opinion. An elaborate philological study by Dr. J.H. Trumbull[86] of the numerals used by many of the North American Indian tribes reveals the presence in the languages of these tribes of a few, but only a few, finger names which are used without change as numeral expressions also. Sometimes the finger gives a name not its own to the numeral with which it is associated in counting—as in the Chippeway dialect, which has nawi-nindj, middle of the hand, and nisswi, 3; and the Cheyenne, where notoyos, middle finger, and na-nohhtu, 8, are closely related. In other parts of the world isolated examples of the transference of finger names to numerals are also found. Of these a well-known example is furnished by the Zulu numerals, where "tatisitupa, taking the thumb, becomes a numeral for six. Then the verb komba, to point, indicating the forefinger, or 'pointer,' makes the next numeral, seven. Thus, answering the question, 'How much did your master give you?' a Zulu would say, 'U kombile,' 'He pointed with his forefinger,' i.e. 'He gave me seven'; and this curious way of using the numeral verb is also shown in such an example as 'amahasi akombile,' 'the horses have pointed,' i.e. 'there were seven of them.' In like manner, Kijangalobili, 'keep back two fingers,' i.e. eight, and Kijangalolunje, 'keep back one finger,' i.e. nine, lead on to kumi, ten."[87]

Returning for a moment to the consideration of number systems in the formation of which the influence of the hand has been paramount, we find still further variations of the method already noticed of constructing names for the fives, tens, and twenties, as well as for the intermediate numbers. Instead of the simple words "hand," "foot," etc., we not infrequently meet with some paraphrase for one or for all these terms, the derivation of which is unmistakable. The Nengones,[88] an island tribe of the Indian Ocean, though using the word "man" for 20, do not employ explicit hand or foot words, but count

1. sa. 2. rewe. 3. tini. 4. etse. 5. se dono = the end (of the first hand). 6. dono ne sa = end and 1. 7. dono ne rewe = end and 2. 8. dono ne tini = end and 3. 9. dono ne etse = end and 4. 10. rewe tubenine = 2 series (of fingers). 11. rewe tubenine ne sa re tsemene = 2 series and 1 on the next? 20. sa re nome = 1 man. 30. sa re nome ne rewe tubenine = 1 man and 2 series. 40. rewe ne nome = 2 men.

Examples like the above are not infrequent. The Aztecs used for 10 the word matlactli, hand-half, i.e. the hand half of a man, and for 20 cempoalli, one counting.[89] The Point Barrow Eskimos call 10 kodlin, the upper part, i.e. of a man. One of the Ewe dialects of Western Africa[90] has ewo, done, for 10; while, curiously enough, 9, asieke, is a digital word, meaning "to part (from) the hand."

In numerous instances also some characteristic word not of hand derivation is found, like the Yoruba ogodzi, string, which becomes a numeral for 40, because 40 cowries made a "string"; and the Maori tekau, bunch, which signifies 10. The origin of this seems to have been the custom of counting yams and fish by "bunches" of ten each.[91]

Another method of forming numeral words above 5 or 10 is found in the presence of such expressions as second 1, second 2, etc. In languages of rude construction and incomplete development the simple numeral scale is often found to end with 5, and all succeeding numerals to be formed from the first 5. The progression from that point may be 5-1, 5-2, etc., as in the numerous quinary scales to be noticed later, or it may be second 1, second 2, etc., as in the Niam Niam dialect of Central Africa, where the scale is[92]

1. sa. 2. uwi. 3. biata. 4. biama. 5. biswi. 6. batissa = 2d 1. 7. batiwwi = 2d 2. 8. batti-biata = 2d 3. 9. batti-biama = 2d 4. 10. bauwe = 2d 5.

That this method of progression is not confined to the least developed languages, however, is shown by a most cursory examination of the numerals of our American Indian tribes, where numeral formation like that exhibited above is exceedingly common. In the Kootenay dialect,[93] of British Columbia, qaetsa, 4, and wo-qaetsa, 8, are obviously related, the latter word probably meaning a second 4. Most of the native languages of British Columbia form their words for 7 and 8 from those which signify 2 and 3; as, for example, the Heiltsuk,[94] which shows in the following words a most obvious correspondence:

2. matl. 7. matlaaus. 3. yutq. 8. yutquaus.

In the Choctaw language[95] the relation between 2 and 7, and 3 and 8, is no less clear. Here the words are:

2. tuklo. 7. untuklo. 3. tuchina. 8. untuchina.

The Nez Perces[96] repeat the first three words of their scale in their 6, 7, and 8 respectively, as a comparison of these numerals will show.

1. naks. 6. oilaks. 2. lapit. 7. oinapt. 3. mitat. 8. oimatat.

In all these cases the essential point of the method is contained in the repetition, in one way or another, of the numerals of the second quinate, without the use with each one of the word for 5. This may make 6, 7, 8, and 9 appear as second 1, second 2, etc., or another 1, another 2, etc.; or, more simply still, as 1 more, 2 more, etc. It is the method which was briefly discussed in the early part of the present chapter, and is by no means uncommon. In a decimal scale this repetition would begin with 11 instead of 6; as in the system found in use in Tagala and Pampanaga, two of the Philippine Islands, where, for example, 11, 12, and 13 are:[97]

11. labi-n-isa = over 1. 12. labi-n-dalaua = over 2. 13. labi-n-tatlo = over 3.

A precisely similar method of numeral building is used by some of our Western Indian tribes. Selecting a few of the Assiniboine numerals[98] as an illustration, we have

11. ak kai washe = more 1. 12. ak kai noom pah = more 2. 13. ak kai yam me nee = more 3. 14. ak kai to pah = more 4. 15. ak kai zap tah = more 5. 16. ak kai shak pah = more 6, etc.

A still more primitive structure is shown in the numerals of the Mboushas[99] of Equatorial Africa. Instead of using 5-1, 5-2, 5-3, 5-4, or 2d 1, 2d 2, 2d 3, 2d 4, in forming their numerals from 6 to 9, they proceed in the following remarkable and, at first thought, inexplicable manner to form their compound numerals:

1. ivoco. 2. beba. 3. belalo. 4. benai. 5. betano. 6. ivoco beba = 1-2. 7. ivoco belalo = 1-3. 8. ivoco benai = 1-4. 9. ivoco betano = 1-5. 10. dioum.

No explanation is given by Mr. du Chaillu for such an apparently incomprehensible form of expression as, for example, 1-3, for 7. Some peculiar finger pantomime may accompany the counting, which, were it known, would enlighten us on the Mbousha's method of arriving at so anomalous a scale. Mere repetition in the second quinate of the words used in the first might readily be explained by supposing the use of fingers absolutely indispensable as an aid to counting, and that a certain word would have one meaning when associated with a certain finger of the left hand, and another meaning when associated with one of the fingers of the right. Such scales are, if the following are correct, actually in existence among the islands of the Pacific.

BALAD.[100] UEA.[100]

1. parai. 1. tahi. 2. paroo. 2. lua. 3. pargen. 3. tolu. 4. parbai. 4. fa. 5. panim. 5. lima. 6. parai. 6. tahi. 7. paroo. 7. lua. 8. pargen. 8. tolu. 9. parbai. 9. fa. 10. panim. 10. lima.

Such examples are, I believe, entirely unique among primitive number systems.

In numeral scales where the formative process has been of the general nature just exhibited, irregularities of various kinds are of frequent occurrence. Hand numerals may appear, and then suddenly disappear, just where we should look for them with the greatest degree of certainty. In the Ende,[101] a dialect of the Flores Islands, 5, 6, and 7 are of hand formation, while 8 and 9 are of entirely different origin, as the scale shows.

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

One special point to be noticed in this scale is the irregularity that prevails between 7, 8, 9. The formation of 7 is of the most ordinary kind; 8 is 2 fours—common enough duplication; while 9 appears to be 10 - 1. All of these modes of compounding are, in their own way, regular; but the irregularity consists in using all three of them in connective numerals in the same system. But, odd as this jumble seems, it is more than matched by that found in the scale of the Karankawa Indians,[102] an extinct tribe formerly inhabiting the coast region of Texas. The first ten numerals of this singular array are:

1. natsa. 2. haikia. 3. kachayi. 4. hayo hakn = 2 x 2. 5. natsa behema = 1 father, i.e. of the fingers. 6. hayo haikia = 3 x 2? 7. haikia natsa = 2 + 5? 8. haikia behema = 2 fathers? 9. haikia doatn = 2d from 10? 10. doatn habe.

Systems like the above, where chaos instead of order seems to be the ruling principle, are of occasional occurrence, but they are decidedly the exception.

In some of the cases that have been adduced for illustration it is to be noticed that the process of combination begins with 7 instead of with 6. Among others, the scale of the Pigmies of Central Africa[103] and that of the Mosquitos[104] of Central America show this tendency. In the Pigmy scale the words for 1 and 6 are so closely akin that one cannot resist the impression that 6 was to them a new 1, and was thus named.

MOSQUITO. PIGMY.

1. kumi. ujju. 2. wal. ibari. 3. niupa. ikaro. 4. wal-wal = 2-2. ikwanganya. 5. mata-sip = fingers of 1 hand. bumuti. 6. matlalkabe. ijju. 7. matlalkabe pura kumi = 6 and 1. bumutti-na-ibali = 5 and 2. 8. matlalkabe pura wal = 6 and 2. bumutti-na-ikaro = 5 and 3. 9. matlalkabe pura niupa = 6 and 3. bumutti-na-ikwanganya = 5 and 4. 10. mata wal sip = fingers of 2 hands. mabo = half man.

The Mosquito scale is quite exceptional in forming 7, 8, and 9 from 6, instead of from 5. The usual method, where combinations appear between 6 and 10, is exhibited by the Pigmy scale. Still another species of numeral form, quite different from any that have already been noticed, is found in the Yoruba[105] scale, which is in many respects one of the most peculiar in existence. Here the words for 11, 12, etc., are formed by adding the suffix -la, great, to the words for 1, 2, etc., thus:

1. eni, or okan. 2. edzi. 3. eta. 4. erin. 5. arun. 6. efa. 7. edze. 8. edzo. 9. esan. 10. ewa. 11. okanla = great 1. 12. edzila = great 2. 13. etala = great 3. 14. erinla = great 4, etc. 40. ogodzi = string. 200. igba = heap.

The word for 40 was adopted because cowrie shells, which are used for counting, were strung by forties; and igba, 200, because a heap of 200 shells was five strings, and thus formed a convenient higher unit for reckoning. Proceeding in this curious manner,[106] they called 50 strings 1 afo or head; and to illustrate their singular mode of reckoning—the king of the Dahomans, having made war on the Yorubans, and attacked their army, was repulsed and defeated with a loss of "two heads, twenty strings, and twenty cowries" of men, or 4820.

The number scale of the Abipones,[107] one of the low tribes of the Paraguay region, contains two genuine curiosities, and by reason of those it deserves a place among any collection of numeral scales designed to exhibit the formation of this class of words. It is:

1. initara = 1 alone. 2. inoaka. 3. inoaka yekaini = 2 and 1. 4. geyenknate = toes of an ostrich. 5. neenhalek = a five coloured, spotted hide, or hanambegen = fingers of 1 hand. 10. lanamrihegem = fingers of both hands. 20. lanamrihegem cat gracherhaka anamichirihegem = fingers of both hands together with toes of both feet.

That the number sense of the Abipones is but little, if at all, above that of the native Australian tribes, is shown by their expressing 3 by the combination 2 and 1. This limitation, as we have already seen, is shared by the Botocudos, the Chiquitos, and many of the other native races of South America. But the Abipones, in seeking for words with which to enable themselves to pass beyond the limit 3, invented the singular terms just given for 4 and 5. The ostrich, having three toes in front and one behind on each foot presented them with a living example of 3 + 1; hence "toes of an ostrich" became their numeral for 4. Similarly, the number of colours in a certain hide being five, the name for that hide was adopted as their next numeral. At this point they began to resort to digital numeration also; and any higher number is expressed by that method.

In the sense in which the word is defined by mathematicians, number is a pure, abstract concept. But a moment's reflection will show that, as it originates among savage races, number is, and from the limitations of their intellect must be, entirely concrete. An abstract conception is something quite foreign to the essentially primitive mind, as missionaries and explorers have found to their chagrin. The savage can form no mental concept of what civilized man means by such a word as "soul"; nor would his idea of the abstract number 5 be much clearer. When he says five, he uses, in many cases at least, the same word that serves him when he wishes to say hand; and his mental concept when he says five is of a hand. The concrete idea of a closed fist or an open hand with outstretched fingers, is what is upper-most in his mind. He knows no more and cares no more about the pure number 5 than he does about the law of the conservation of energy. He sees in his mental picture only the real, material image, and his only comprehension of the number is, "these objects are as many as the fingers on my hand." Then, in the lapse of the long interval of centuries which intervene between lowest barbarism and highest civilization, the abstract and the concrete become slowly dissociated, the one from the other. First the actual hand picture fades away, and the number is recognized without the original assistance furnished by the derivation of the word. But the number is still for a long time a certain number of objects, and not an independent concept. It is only when the savage ceases to be wholly an animal, and becomes a thinking human being, that number in the abstract can come within the grasp of his mind. It is at this point that mere reckoning ceases, and arithmetic begins.



CHAPTER IV.

THE ORIGIN OF NUMBER WORDS. (CONTINUED.)

By the slow, and often painful, process incident to the extension and development of any mental conception in a mind wholly unused to abstractions, the savage gropes his way onward in his counting from 1, or more probably from 2, to the various higher numbers required to form his scale. The perception of unity offers no difficulty to his mind, though he is conscious at first of the object itself rather than of any idea of number associated with it. The concept of duality, also, is grasped with perfect readiness. This concept is, in its simplest form, presented to the mind as soon as the individual distinguishes himself from another person, though the idea is still essentially concrete. Perhaps the first glimmering of any real number thought in connection with 2 comes when the savage contrasts one single object with another—or, in other words, when he first recognizes the pair. At first the individuals composing the pair are simply "this one," and "that one," or "this and that"; and his number system now halts for a time at the stage when he can, rudely enough it may be, count 1, 2, many. There are certain cases where the forms of 1 and 2 are so similar than one may readily imagine that these numbers really were "this" and "that" in the savage's original conception of them; and the same likeness also occurs in the words for 3 and 4, which may readily enough have been a second "this" and a second "that." In the Lushu tongue the words for 1 and 2 are tizi and tazi respectively. In Koriak we find ngroka, 3, and ngraka, 4; in Kolyma, niyokh, 3, and niyakh, 4; and in Kamtschatkan, tsuk, 3, and tsaak, 4.[108] Sometimes, as in the case of the Australian races, the entire extent of the count is carried through by means of pairs. But the natural theory one would form is, that 2 is the halting place for a very long time; that up to this point the fingers may or may not have been used—probably not; and that when the next start is made, and 3, 4, 5, and so on are counted, the fingers first come into requisition. If the grammatical structure of the earlier languages of the world's history is examined, the student is struck with the prevalence of the dual number in them—something which tends to disappear as language undergoes extended development. The dual number points unequivocally to the time when 1 and 2 were the numbers at mankind's disposal; to the time when his three numeral concepts, 1, 2, many, each demanded distinct expression. With increasing knowledge the necessity for this differentiatuin would pass away, and but two numbers, singular and plural, would remain. Incidentally it is to be noticed that the Indo-European words for 3—three, trois, drei, tres, tri, etc., have the same root as the Latin trans, beyond, and give us a hint of the time when our Aryan ancestors counted in the manner I have just described.

The first real difficulty which the savage experiences in counting, the difficulty which comes when he attempts to pass beyond 2, and to count 3, 4, and 5, is of course but slight; and these numbers are commonly used and readily understood by almost all tribes, no matter how deeply sunk in barbarism we find them. But the instances that have already been cited must not be forgotten. The Chiquitos do not, in their primitive state, properly count at all; the Andamans, the Veddas, and many of the Australian tribes have no numerals higher than 2; others of the Australians and many of the South Americans stop with 3 or 4; and tribes which make 5 their limit are still more numerous. Hence it is safe to assert that even this insignificant number is not always reached with perfect ease. Beyond 5 primitive man often proceeds with the greatest difficulty. Most savages, even those of the tribes just mentioned, can really count above here, even though they have no words with which to express their thought. But they do it with reluctance, and as they go on they quickly lose all sense of accuracy. This has already been commented on, but to emphasize it afresh the well-known example given by Mr. Oldfield from his own experience among the Watchandies may be quoted.[109] "I once wished to ascertain the exact number of natives who had been slain on a certain occasion. The individual of whom I made the inquiry began to think over the names ... assigning one of his fingers to each, and it was not until after many failures, and consequent fresh starts, that he was able to express so high a number, which he at length did by holding up his hand three times, thus giving me to understand that fifteen was the answer to this most difficult arithmetical question." This meagreness of knowledge in all things pertaining to numbers is often found to be sharply emphasized in the names adopted by savages for their numeral words. While discussing in a previous chapter the limits of number systems, we found many instances where anything above 2 or 3 was designated by some one of the comprehensive terms much, many, very many; these words, or such equivalents as lot, heap, or plenty, serving as an aid to the finger pantomime necessary to indicate numbers for which they have no real names. The low degree of intelligence and civilization revealed by such words is brought quite as sharply into prominence by the word occasionally found for 5. Whenever the fingers and hands are used at all, it would seem natural to expect for 5 some general expression signifying hand, for 10 both hands, and for 20 man. Such is, as we have already seen, the ordinary method of progression, but it is not universal. A drop in the scale of civilization takes us to a point where 10, instead of 20, becomes the whole man. The Kusaies,[110] of Strong's Island, call 10 sie-nul, 1 man, 30 tol-nul, 3 men, 40 a naul, 4 men, etc.; and the Ku-Mbutti[111] of central Africa have mukko, 10, and moku, man. If 10 is to be expressed by reference to the man, instead of his hands, it might appear more natural to employ some such expression as that adopted by the African Pigmies,[112] who call 10 mabo, and man mabo-mabo. With them, then, 10 is perhaps "half a man," as it actually is among the Towkas of South America; and we have already seen that with the Aztecs it was matlactli, the "hand half" of a man.[113] The same idea crops out in the expression used by the Nicobar Islanders for 30—heam-umdjome ruktei, 1 man (and a) half.[114] Such nomenclature is entirely natural, and it accords with the analogy offered by other words of frequent occurrence in the numeral scales of savage races. Still, to find 10 expressed by the term man always conveys an impression of mental poverty; though it may, of course, be urged that this might arise from the fact that some races never use the toes in counting, but go over the fingers again, or perhaps bring into requisition the fingers of a second man to express the second 10. It is not safe to postulate an extremely low degree of civilization from the presence of certain peculiarities of numeral formation. Only the most general statements can be ventured on, and these are always subject to modification through some circumstance connected with environment, mode of living, or intercourse with other tribes. Two South American races may be cited, which seem in this respect to give unmistakable evidence of being sunk in deepest barbarism. These are the Juri and the Cayriri, who use the same word for man and for 5. The former express 5 by ghomen apa, 1 man,[115] and the latter by ibicho, person.[116] The Tasmanians of Oyster Bay use the native word of similar meaning, puggana, man,[117] for 5.

Wherever the numeral 20 is expressed by the term man, it may be expected that 40 will be 2 men, 60, 3 men, etc. This form of numeration is usually, though not always, carried as far as the system extends; and it sometimes leads to curious terms, of which a single illustration will suffice. The San Blas Indians, like almost all the other Central and South American tribes, count by digit numerals, and form their twenties as follows:[118]

20. tula guena = man 1. 40. tula pogua = man 2. 100. tula atala = man 5. 120. tula nergua = man 6. 1000. tula wala guena = great 1 man.

The last expression may, perhaps, be translated "great hundred," though the literal meaning is the one given. If 10, instead of 20, is expressed by the word "man," the multiples of 10 follow the law just given for multiples of 20. This is sufficiently indicated by the Kusaie scale; or equally well by the Api words for 100 and 200, which are[119]

duulimo toromomo = 10 times the whole man.

duulimo toromomo va juo = 10 times the whole man taken 2 times.

As an illustration of the legitimate result which is produced by the attempt to express high numbers in this manner the term applied by educated native Greenlanders[120] for a thousand may be cited. This numeral, which is, of course, not in common use, is

inuit kulit tatdlima nik kuleriartut navdlugit = 10 men 5 times 10 times come to an end.

It is worth noting that the word "great," which appears in the scale of the San Blas Indians, is not infrequently made use of in the formation of higher numeral words. The African Mabas[121] call 10 atuk, great 1; the Hottentots[122] and the Hidatsa Indians call 100 great 10, their words being gei disi and pitikitstia respectively.

The Nicaraguans[123] express 100 by guhamba, great 10, and 400 by dinoamba, great 20; and our own familiar word "million," which so many modern languages have borrowed from the Italian, is nothing more nor less than a derivative of the Latin mille, and really means "great thousand." The Dakota[124] language shows the same origin for its expression of 1,000,000, which is kick ta opong wa tunkah, great 1000. The origin of such terms can hardly be ascribed to poverty of language. It is found, rather, in the mental association of the larger with the smaller unit, and the consequent repetition of the name of the smaller. Any unit, whether it be a single thing, a dozen, a score, a hundred, a thousand, or any other unit, is, whenever used, a single and complete group; and where the relation between them is sufficiently close, as in our "gross" and "great gross," this form of nomenclature is natural enough to render it a matter of some surprise that it has not been employed more frequently. An old English nursery rhyme makes use of this association, only in a manner precisely the reverse of that which appears now and then in numeral terms. In the latter case the process is always one of enlargement, and the associative word is "great." In the following rhyme, constructed by the mature for the amusement of the childish mind, the process is one of diminution, and the associative word is "little":

One's none, Two's some, Three's a many, Four's a penny, Five's a little hundred.[125]

Any real numeral formation by the use of "little," with the name of some higher unit, would, of course, be impossible. The numeral scale must be complete before the nursery rhyme can be manufactured.

It is not to be supposed from the observations that have been made on the formation of savage numeral scales that all, or even the majority of tribes, proceed in the awkward and faltering manner indicated by many of the examples quoted. Some of the North American Indian tribes have numeral scales which are, as far as they go, as regular and almost as simple as our own. But where digital numeration is extensively resorted to, the expressions for higher numbers are likely to become complex, and to act as a real bar to the extension of the system. The same thing is true, to an even greater degree, of tribes whose number sense is so defective that they begin almost from the outset to use combinations. If a savage expresses the number 3 by the combination 2-1, it will at once be suspected that his numerals will, by the time he reaches 10 or 20, become so complex and confused that numbers as high as these will be expressed by finger pantomime rather than by words. Such is often the case; and the comment is frequently made by explorers that the tribes they have visited have no words for numbers higher than 3, 4, 5, 10, or 20, but that counting is carried beyond that point by the aid of fingers or other objects. So reluctant, in many cases, are savages to count by words, that limits have been assigned for spoken numerals, which subsequent investigation proved to fall far short of the real extent of the number systems to which they belonged. One of the south-western Indian tribes of the United States, the Comanches, was for a time supposed to have no numeral words below 10, but to count solely by the use of fingers. But the entire scale of this taciturn tribe was afterward discovered and published.

To illustrate the awkward and inconvenient forms of expression which abound in primitive numeral nomenclature, one has only to draw from such scales as those of the Zuni, or the Point Barrow Eskimos, given in the last chapter. Terms such as are found there may readily be duplicated from almost any quarter of the globe. The Soussous of Sierra Leone[126] call 99 tongo solo manani nun solo manani, i.e. to take (10 understood) 5 + 4 times and 5 + 4. The Malagasy expression for 1832 is[127] roambistelo polo amby valonjato amby arivo, 2 + 30 + 800 + 1000. The Aztec equivalent for 399 is[128] caxtolli onnauh poalli ipan caxtolli onnaui, (15 + 4) x 20 + 15 + 4; and the Sioux require for 29 the ponderous combination[129] wick a chimen ne nompah sam pah nep e chu wink a. These terms, long and awkward as they seem, are only the legitimate results which arise from combining the names of the higher and lower numbers, according to the peculiar genius of each language. From some of the Australian tribes are derived expressions still more complex, as for 6, marh-jin-bang-ga-gudjir-gyn, half the hands and 1; and for 15, marh-jin-belli-belli-gudjir-jina-bang-ga, the hand on either side and half the feet.[130] The Mare tribe, one of the numerous island tribes of Melanesia,[131] required for a translation of the numeral 38, which occurs in John v. 5, "had an infirmity thirty and eight years," the circumlocution, "one man and both sides five and three." Such expressions, curious as they seem at first thought, are no more than the natural outgrowth of systems built up by the slow and tedious process which so often obtains among primitive races, where digit numerals are combined in an almost endless variety of ways, and where mere reduplication often serves in place of any independent names for higher units. To what extent this may be carried is shown by the language of the Cayubabi,[132] who have for 10 the word tunca, and for 100 and 1000 the compounds tunca tunca, and tunca tunca tunca respectively; or of the Sapibocones, who call 10 bururuche, hand hand, and 100 buruche buruche, hand hand hand hand.[133] More remarkable still is the Ojibwa language, which continues its numeral scale without limit, furnishing combinations which are really remarkable; as, e.g., that for 1,000,000,000, which is me das wac me das wac as he me das wac,[134] 1000 x 1000 x 1000. The Winnebago expression for the same number,[135] ho ke he hhuta hhu chen a ho ke he ka ra pa ne za is no less formidable, but it has every appearance of being an honest, native combination. All such primitive terms for larger numbers must, however, be received with caution. Savages are sometimes eager to display a knowledge they do not possess, and have been known to invent numeral words on the spot for the sake of carrying their scales to as high a limit as possible. The Choctaw words for million and billion are obvious attempts to incorporate the corresponding English terms into their own language.[136] For million they gave the vocabulary-hunter the phrase mil yan chuffa, and for billion, bil yan chuffa. The word chuffa signifies 1, hence these expressions are seen at a glance to be coined solely for the purpose of gratifying a little harmless Choctaw vanity. But this is innocence itself compared with the fraud perpetrated on Labillardiere by the Tonga Islanders, who supplied the astonished and delighted investigator with a numeral vocabulary up to quadrillions. Their real limit was afterward found to be 100,000, and above that point they had palmed off as numerals a tolerably complete list of the obscene words of their language, together with a few nonsense terms. These were all accepted and printed in good faith, and the humiliating truth was not discovered until years afterward.[137]

One noteworthy and interesting fact relating to numeral nomenclature is the variation in form which words of this class undergo when applied to different classes of objects. To one accustomed as we are to absolute and unvarying forms for numerals, this seems at first a novel and almost unaccountable linguistic freak. But it is not uncommon among uncivilized races, and is extensively employed by so highly enlightened a people, even, as the Japanese. This variation in form is in no way analogous to that produced by inflectional changes, such as occur in Hebrew, Greek, Latin, etc. It is sufficient in many cases to produce almost an entire change in the form of the word; or to result in compounds which require close scrutiny for the detection of the original root. For example, in the Carrier, one of the Dene dialects of western Canada, the word tha means 3 things; thane, 3 persons; that, 3 times; thatoen, in 3 places; thauh, in 3 ways; thailtoh, all of the 3 things; thahoeltoh, all of the 3 persons; and thahultoh, all of the 3 times.[138] In the Tsimshian language of British Columbia we find seven distinct sets of numerals "which are used for various classes of objects that are counted. The first set is used in counting where there is no definite object referred to; the second class is used for counting flat objects and animals; the third for counting round objects and divisions of time; the fourth for counting men; the fifth for counting long objects, the numerals being composed with kan, tree; the sixth for counting canoes; and the seventh for measures. The last seem to be composed with anon, hand."[139] The first ten numerals of each of these classes is given in the following table:

+ + -+ -+ -+ + + -+ -+ No. Counting Flat Round Men Long Canoes Measures Objects Objects Objects + + -+ -+ -+ + + -+ -+ 1 gyak gak g'erel k'al k'awutskan k'amaet k'al 2 t'epqat t'epqat goupel t'epqadal gaopskan g'alpēeltk gulbel 3 guant guant gutle gulal galtskan galtskantk guleont 4 tqalpq tqalpq tqalpq tqalpqdal tqaapskan tqalpqsk tqalpqalont 5 kctōnc kctōnc kctōnc kcenecal k'etoentskan kctōonsk kctonsilont 6 k'alt k'alt k'alt k'aldal k'aoltskan k'altk k'aldelont 7 t'epqalt t'epqalt t'epqalt t'epqaldal t'epqaltskan t'epqaltk t'epqaldelont 8 guandalt yuktalt yuktalt yuktleadal ek'tlaedskan yuktaltk yuktaldelont 9 kctemac kctemac kctemac kctemacal kctemaestkan kctemack kctemasilont 10 gy'ap gy'ap kpēel kpal kpēetskan gy'apsk kpeont + + -+ -+ -+ + + -+ -+

Remarkable as this list may appear, it is by no means as extensive as that derived from many of the other British Columbian tribes. The numerals of the Shushwap, Stlatlumh, Okanaken, and other languages of this region exist in several different forms, and can also be modified by any of the innumerable suffixes of these tongues.[140] To illustrate the almost illimitable number of sets that may be formed, a table is given of "a few classes, taken from the Heiltsuk dialect.[141] It appears from these examples that the number of classes is unlimited."

+ -+ -+ + + One. Two. Three. + -+ -+ + + Animate. menok maalok yutuk Round. menskam masem yutqsem Long. ments'ak mats'ak yututs'ak Flat. menaqsa matlqsa yutqsa Day. op'enequls matlp'enequls yutqp'enequls Fathom. op'enkh matlp'enkh yutqp'enkh Grouped together. matloutl yutoutl Groups of objects. nemtsmots'utl matltsmots'utl yutqtsmots'utl Filled cup. menqtlala matl'aqtlala yutqtlala Empty cup. menqtla matl'aqtla yutqtla Full box. menskamala masemala yutqsemala Empty box. menskam masem yutqsem Loaded canoe. mentsake mats'ake yututs'ake Canoe with crew. ments'akis mats'akla yututs'akla Together on beach. maalis Together in house, etc. maalitl + -+ -+ + +

Variation in numeral forms such as is exhibited in the above tables is not confined to any one quarter of the globe; but it is more universal among the British Columbian Indians than among any other race, and it is a more characteristic linguistic peculiarity of this than of any other region, either in the Old World or in the New. It was to some extent employed by the Aztecs,[142] and its use is current among the Japanese; in whose language Crawfurd finds fourteen different classes of numerals "without exhausting the list."[143]

In examining the numerals of different languages it will be found that the tens of any ordinary decimal scale are formed in the same manner as in English. Twenty is simply 2 times 10; 30 is 3 times 10, and so on. The word "times" is, of course, not expressed, any more than in English; but the expressions briefly are, 2 tens, 3 tens, etc. But a singular exception to this method is presented by the Hebrew, and other of the Semitic languages. In Hebrew the word for 20 is the plural of the word for 10; and 30, 40, 50, etc. to 90 are plurals of 3, 4, 5, 6, 7, 8, 9. These numerals are as follows:[144]

10, eser, 20, eserim, 3, shalosh, 30, shaloshim, 4, arba, 40, arbaim, 5, chamesh, 50, chamishshim, 6, shesh, 60, sheshshim, 7, sheba, 70, shibim, 8, shemoneh 80, shemonim, 9, tesha, 90, tishim.

The same formation appears in the numerals of the ancient Phoenicians,[145] and seems, indeed, to be a well-marked characteristic of the various branches of this division of the Caucasian race. An analogous method appears in the formation of the tens in the Bisayan,[146] one of the Malay numeral scales, where 30, 40, ... 90, are constructed from 3, 4, ... 9, by adding the termination -an.

No more interesting contribution has ever been made to the literature of numeral nomenclature than that in which Dr. Trumbull embodies the results of his scholarly research among the languages of the native Indian tribes of this country.[147] As might be expected, we are everywhere confronted with a digital origin, direct or indirect, in the great body of the words examined. But it is clearly shown that such a derivation cannot be established for all numerals; and evidence collected by the most recent research fully substantiates the position taken by Dr. Trumbull. Nearly all the derivations established are such as to remind us of the meanings we have already seen recurring in one form or another in language after language. Five is the end of the finger count on one hand—as, the Micmac nan, and Mohegan nunon, gone, or spent; the Pawnee sihuks, hands half; the Dakota zaptan, hand turned down; and the Massachusetts napanna, on one side. Ten is the end of the finger count, but is not always expressed by the "both hands" formula so commonly met with. The Cree term for this number is mitatat, no further; and the corresponding word in Delaware is m'tellen, no more. The Dakota 10 is, like its 5, a straightening out of the fingers which have been turned over in counting, or wickchemna, spread out unbent. The same is true of the Hidatsa pitika, which signifies a smoothing out, or straightening. The Pawnee 4, skitiks, is unusual, signifying as it does "all the fingers," or more properly, "the fingers of the hand." The same meaning attaches to this numeral in a few other languages also, and reminds one of the habit some people have of beginning to count on the forefinger and proceeding from there to the little finger. Can this have been the habit of the tribes in question? A suggestion of the same nature is made by the Illinois and Miami words for 8, parare and polane, which signify "nearly ended." Six is almost always digital in origin, though the derivation may be indirect, as in the Illinois kakatchui, passing beyond the middle; and the Dakota shakpe, 1 in addition. Some of these significations are well matched by numerals from the Ewe scales of western Africa, where we find the following:[148]

1. de = a going, i.e. a beginning. (Cf. the Zuni toepinte, taken to start with.) 3. eto = the father (from the middle, or longest finger). 6. ade = the other going. 9. asieke = parting with the hands. 10. ewo = done.

In studying the names for 2 we are at once led away from a strictly digital origin for the terms by which this number is expressed. These names seem to come from four different sources: (1) roots denoting separation or distinction; (2) likeness, equality, or opposition; (3) addition, i.e. putting to, or putting with; (4) coupling, pairing, or matching. They are often related to, and perhaps derived from, names of natural pairs, as feet, hands, eyes, arms, or wings. In the Dakota and Algonkin dialects 2 is almost always related to "arms" or "hands," and in the Athapaskan to "feet." But the relationship is that of common origin, rather than of derivation from these pair-names. In the Puri and Hottentot languages, 2 and "hand" are closely allied; while in Sanskrit, 2 may be expressed by any one of the words kara, hand, bahu, arm, paksha, wing, or netra, eye.[149] Still more remote from anything digital in their derivation are the following, taken at random from a very great number of examples that might be cited to illustrate this point. The Assiniboines call 7, shak ko we, or u she nah, the odd number.[150] The Crow 1, hamat, signifies "the least";[151] the Mississaga 1, pecik, a very small thing.[152] In Javanese, Malay, and Manadu, the words for 1, which are respectively siji, satu, and sabuah, signify 1 seed, 1 pebble, and 1 fruit respectively[153]—words as natural and as much to be expected at the beginning of a number scale as any finger name could possibly be. Among almost all savage races one form or another of palpable arithmetic is found, such as counting by seeds, pebbles, shells, notches, or knots; and the derivation of number words from these sources can constitute no ground for surprise. The Marquesan word for 4 is pona, knot, from the practice of tying breadfruit in knots of 4. The Maori 10 is tekau, bunch, or parcel, from the counting of yams and fish by parcels of 10.[154] The Javanese call 25, lawe, a thread, or string; 50, ekat, a skein of thread; 400, samas, a bit of gold; 800, domas, 2 bits of gold.[155] The Macassar and Butong term for 100 is bilangan, 1 tale or reckoning.[156] The Aztec 20 is cem pohualli, 1 count; 400 is centzontli, 1 hair of the head; and 8000 is xiquipilli, sack.[157] This sack was of such a size as to contain 8000 cacao nibs, or grains, hence the derivation of the word in its numeral sense is perfectly natural. In Japanese we find a large number of terms which, as applied to the different units of the number scale, seem almost purely fanciful. These words, with their meanings as given by a Japanese lexicon, are as follows:

10,000, or 10^4, maen = enormous number. 10^8, oku = a compound of the words "man" and "mind." 10^12, chio = indication, or symptom. 10^16, kei = capital city. 10^20, si = a term referring to grains. 10^24, owi = —— 10^28, jio = extent of land. 10^32, ko = canal. 10^36, kan = some kind of a body of water. 10^40, sai = justice. 10^44, sā = support. 10^48, kioku = limit, or more strictly, ultimate. .01^2, rin = —— .01^3, mo = hair (of some animal). .01^4, shi = thread.

In addition to these, some of the lower fractional values are described by words meaning "very small," "very fine thread," "sand grain," "dust," and "very vague." Taken altogether, the Japanese number system is the most remarkable I have ever examined, in the extent and variety of the higher numerals with well-defined descriptive names. Most of the terms employed are such as to defy any attempt to trace the process of reasoning which led to their adoption. It is not improbable that the choice was, in some of these cases at least, either accidental or arbitrary; but still, the changes in word meanings which occur with the lapse of time may have differentiated significations originally alike, until no trace of kinship would appear to the casual observer. Our numerals "score" and "gross" are never thought of as having any original relation to what is conveyed by the other meanings which attach to these words. But the origin of each, which is easily traced, shows that, in the beginning, there existed a well-defined reason for the selection of these, rather than other terms, for the numbers they now describe. Possibly these remarkable Japanese terms may be accounted for in the same way, though the supposition is, for some reasons, quite improbable. The same may be said for the Malagasy 1000, alina, which also means "night," and the Hebrew 6, shesh, which has the additional signification "white marble," and the stray exceptions which now and then come to the light in this or that language. Such terms as these may admit of some logical explanation, but for the great mass of numerals whose primitive meanings can be traced at all, no explanation whatever is needed; the words are self-explanatory, as the examples already cited show.

A few additional examples of natural derivation may still further emphasize the point just discussed. In Bambarese the word for 10, tank, is derived directly from adang, to count.[158] In the language of Mota, one of the islands of Melanesia, 100 is mel nol, used and done with, referring to the leaves of the cycas tree, with which the count had been carried on.[159] In many other Melanesian dialects[160] 100 is rau, a branch or leaf. In the Torres Straits we find the same number expressed by na won, the close; and in Eromanga it is narolim narolim (2 x 5)(2 x 5).[161] This combination deserves remark only because of the involved form which seems to have been required for the expression of so small a number as 100. A compound instead of a simple term for any higher unit is never to be wondered at, so rude are some of the savage methods of expressing number; but "two fives (times) two fives" is certainly remarkable. Some form like that employed by the Nusqually[162] of Puget Sound for 1000, i.e. paduts-subquaetche, ten hundred, is more in accordance with primitive method. But we are equally likely to find such descriptive phrases for this numeral as the dor paka, banyan roots, of the Torres Islands; rau na hai, leaves of a tree, of Vaturana; or udolu, all, of the Fiji Islands. And two curious phrases for 1000 are those of the Banks' Islands, tar mataqelaqela, eye blind thousand, i.e. many beyond count; and of Malanta, warehune huto, opossum's hairs, or idumie one, count the sand.[163]

The native languages of India, Thibet, and portions of the Indian archipelago furnish us with abundant instances of the formation of secondary numeral scales, which were used only for special purposes, and without in any way interfering with the use of the number words already in use. "Thus the scholars of India, ages ago, selected a set of words for a memoria technica, in order to record dates and numbers. These words they chose for reasons which are still in great measure evident; thus 'moon' or 'earth' expressed 1, there being but one of each; 2 might be called 'eye,' 'wing,' 'arm,' 'jaw,' as going in pairs; for 3 they said 'Rama,' 'fire,' or 'quality,' there being considered to be three Ramas, three kinds of fire, three qualities (guna); for 4 were used 'veda,' 'age,' or 'ocean,' there being four of each recognized; 'season' for 6, because they reckoned six seasons; 'sage' or 'vowel,' for 7, from the seven sages and the seven vowels; and so on with higher numbers, 'sun' for 12, because of his twelve annual denominations, or 'zodiac' from his twelve signs, and 'nail' for 20, a word incidentally bringing in finger notation. As Sanskrit is very rich in synonyms, and as even the numerals themselves might be used, it became very easy to draw up phrases or nonsense verses to record series of numbers by this system of artificial memory."[164]

More than enough has been said to show how baseless is the claim that all numeral words are derived, either directly or indirectly, from the names of fingers, hands, or feet. Connected with the origin of each number word there may be some metaphor, which cannot always be distinctly traced; and where the metaphor was born of the hand or of the foot, we inevitably associate it with the practice of finger counting. But races as fond of metaphor and of linguistic embellishment as are those of the East, or as are our American Indians even, might readily resort to some other source than that furnished by the members of the human body, when in want of a term with which to describe the 5, 10, or any other number of the numeral scale they were unconsciously forming. That the first numbers of a numeral scale are usually derived from other sources, we have some reason to believe; but that all above 2, 3, or at most 4, are almost universally of digital origin we must admit. Exception should properly be made of higher units, say 1000 or anything greater, which could not be expected to conform to any law of derivation governing the first few units of a system.

Collecting together and comparing with one another the great mass of terms by which we find any number expressed in different languages, and, while admitting the great diversity of method practised by different tribes, we observe certain resemblances which were not at first supposed to exist. The various meanings of 1, where they can be traced at all, cluster into a little group of significations with which at last we come to associate the idea of unity. Similarly of 2, or 5, or 10, or any one of the little band which does picket duty for the advance guard of the great host of number words which are to follow. A careful examination of the first decade warrants the assertion that the probable meaning of any one of the units will be found in the list given below. The words selected are intended merely to serve as indications of the thought underlying the savage's choice, and not necessarily as the exact term by means of which he describes his number. Only the commonest meanings are included in the tabulation here given.

1 = existence, piece, group, beginning. 2 = repetition, division, natural pair. 3 = collection, many, two-one. 4 = two twos. 5 = hand, group, division, 6 = five-one, two threes, second one. 7 = five-two, second two, three from ten. 8 = five-three, second three, two fours, two from ten. 9 = five-four, three threes, one from ten. 10 = one (group), two fives (hands), half a man, one man. 15 = ten-five, one foot, three fives. 20 = two tens, one man, two feet.[165]



CHAPTER V.

MISCELLANEOUS NUMBER BASES.

In the development and extension of any series of numbers into a systematic arrangement to which the term system may be applied, the first and most indispensable step is the selection of some number which is to serve as a base. When the savage begins the process of counting he invents, one after another, names with which to designate the successive steps of his numerical journey. At first there is no attempt at definiteness in the description he gives of any considerable number. If he cannot show what he means by the use of his fingers, or perhaps by the fingers of a single hand, he unhesitatingly passes it by, calling it many, heap, innumerable, as many as the leaves on the trees, or something else equally expressive and equally indefinite. But the time comes at last when a greater degree of exactness is required. Perhaps the number 11 is to be indicated, and indicated precisely. A fresh mental effort is required of the ignorant child of nature; and the result is "all the fingers and one more," "both hands and one more," "one on another count," or some equivalent circumlocution. If he has an independent word for 10, the result will be simply ten-one. When this step has been taken, the base is established. The savage has, with entire unconsciousness, made all his subsequent progress dependent on the number 10, or, in other words, he has established 10 as the base of his number system. The process just indicated may be gone through with at 5, or at 20, thus giving us a quinary or a vigesimal, or, more probably, a mixed system; and, in rare instances, some other number may serve as the point of departure from simple into compound numeral terms. But the general idea is always the same, and only the details of formation are found to differ.

Without the establishment of some base any system of numbers is impossible. The savage has no means of keeping track of his count unless he can at each step refer himself to some well-defined milestone in his course. If, as has been pointed out in the foregoing chapters, confusion results whenever an attempt is made to count any number which carries him above 10, it must at once appear that progress beyond that point would be rendered many times more difficult if it were not for the fact that, at each new step, he has only to indicate the distance he has progressed beyond his base, and not the distance from his original starting-point. Some idea may, perhaps, be gained of the nature of this difficulty by imagining the numbers of our ordinary scale to be represented, each one by a single symbol different from that used to denote any other number. How long would it take the average intellect to master the first 50 even, so that each number could without hesitation be indicated by its appropriate symbol? After the first 50 were once mastered, what of the next 50? and the next? and the next? and so on. The acquisition of a scale for which we had no other means of expression than that just described would be a matter of the extremest difficulty, and could never, save in the most exceptional circumstances, progress beyond the attainment of a limit of a few hundred. If the various numbers in question were designated by words instead of by symbols, the difficulty of the task would be still further increased. Hence, the establishment of some number as a base is not only a matter of the very highest convenience, but of absolute necessity, if any save the first few numbers are ever to be used.

In the selection of a base,—of a number from which he makes a fresh start, and to which he refers the next steps in his count,—the savage simply follows nature when he chooses 10, or perhaps 5 or 20. But it is a matter of the greatest interest to find that other numbers have, in exceptional cases, been used for this purpose. Two centuries ago the distinguished philosopher and mathematician, Leibnitz, proposed a binary system of numeration. The only symbols needed in such a system would be 0 and 1. The number which is now symbolized by the figure 2 would be represented by 10; while 3, 4, 5, 6, 7, 8, etc., would appear in the binary notation as 11, 100, 101, 110, 111, 1000, etc. The difficulty with such a system is that it rapidly grows cumbersome, requiring the use of so many figures for indicating any number. But Leibnitz found in the representation of all numbers by means of the two digits 0 and 1 a fitting symbolization of the creation out of chaos, or nothing, of the entire universe by the power of the Deity. In commemoration of this invention a medal was struck bearing on the obverse the words

Numero Deus impari gaudet,

and on the reverse,

Omnibus ex nihilo ducendis sufficit Unum.[166]

This curious system seems to have been regarded with the greatest affection by its inventor, who used every endeavour in his power to bring it to the notice of scholars and to urge its claims. But it appears to have been received with entire indifference, and to have been regarded merely as a mathematical curiosity.

Unknown to Leibnitz, however, a binary method of counting actually existed during that age; and it is only at the present time that it is becoming extinct. In Australia, the continent that is unique in its flora, its fauna, and its general topography, we find also this anomaly among methods of counting. The natives, who are to be classed among the lowest and the least intelligent of the aboriginal races of the world, have number systems of the most rudimentary nature, and evince a decided tendency to count by twos. This peculiarity, which was to some extent shared by the Tasmanians, the island tribes of the Torres Straits, and other aboriginal races of that region, has by some writers been regarded as peculiar to their part of the world; as though a binary number system were not to be found elsewhere. This attempt to make out of the rude and unusual method of counting which obtained among the Australians a racial characteristic is hardly justified by fuller investigation. Binary number systems, which are given in full on another page, are found in South America. Some of the Dravidian scales are binary;[167] and the marked preference, not infrequently observed among savage races, for counting by pairs, is in itself a sufficient refutation of this theory. Still it is an unquestionable fact that this binary tendency is more pronounced among the Australians than among any other extensive number of kindred races. They seldom count in words above 4, and almost never as high as 7. One of the most careful observers among them expresses his doubt as to a native's ability to discover the loss of two pins, if he were first shown seven pins in a row, and then two were removed without his knowledge.[168] But he believes that if a single pin were removed from the seven, the Blackfellow would become conscious of its loss. This is due to his habit of counting by pairs, which enables him to discover whether any number within reasonable limit is odd or even. Some of the negro tribes of Africa, and of the Indian tribes of America, have the same habit. Progression by pairs may seem to some tribes as natural as progression by single units. It certainly is not at all rare; and in Australia its influence on spoken number systems is most apparent.

Any number system which passes the limit 10 is reasonably sure to have either a quinary, a decimal, or a vigesimal structure. A binary scale could, as it is developed in primitive languages, hardly extend to 20, or even to 10, without becoming exceedingly cumbersome. A binary scale inevitably suggests a wretchedly low degree of mental development, which stands in the way of the formation of any number scale worthy to be dignified by the name of system. Take, for example, one of the dialects found among the western tribes of the Torres Straits, where, in general, but two numerals are found to exist. In this dialect the method of counting is:[169]

1. urapun. 2. okosa. 3. okosa urapun = 2-1. 4. okosa okosa = 2-2. 5. okosa okosa urapun = 2-2-1. 6. okosa okosa okosa = 2-2-2.

Anything above 6 they call ras, a lot.

For the sake of uniformity we may speak of this as a "system." But in so doing, we give to the legitimate meaning of the word a severe strain. The customs and modes of life of these people are not such as to require the use of any save the scanty list of numbers given above; and their mental poverty prompts them to call 3, the first number above a single pair, 2-1. In the same way, 4 and 6 are respectively 2 pairs and 3 pairs, while 5 is 1 more than 2 pairs. Five objects, however, they sometimes denote by urapuni-getal, 1 hand. A precisely similar condition is found to prevail respecting the arithmetic of all the Australian tribes. In some cases only two numerals are found, and in others three. But in a very great number of the native languages of that continent the count proceeds by pairs, if indeed it proceeds at all. Hence we at once reject the theory that Australian arithmetic, or Australian counting, is essentially peculiar. It is simply a legitimate result, such as might be looked for in any part of the world, of the barbarism in which the races of that quarter of the world were sunk, and in which they were content to live.

The following examples of Australian and Tasmanian number systems show how scanty was the numerical ability possessed by these tribes, and illustrate fully their tendency to count by twos or pairs.

MURRAY RIVER.[170]

1. enea. 2. petcheval. 3. petchevalenea = 2-1. 4. petcheval peteheval = 2-2.

MAROURA.

1. nukee. 2. barkolo. 3. barkolo nuke = 2-1. 4. barkolo barkolo = 2-2.

LAKE KOPPERAMANA.

1. ngerna. 2. mondroo. 3. barkooloo. 4. mondroo mondroo = 2-2.

MORT NOULAR.

1. gamboden. 2. bengeroo. 3. bengeroganmel = 2-1. 4. bengeroovor bengeroo = 2 + 2.

WIMMERA.

1. keyap. 2. pollit. 3. pollit keyap = 2-1. 4. pollit pollit = 2-2.

POPHAM BAY.

1. motu. 2. lawitbari. 3. lawitbari-motu = 2-1.

KAMILAROI.[171]

1. mal. 2. bularr. 3. guliba. 4. bularrbularr = 2-2. 5. bulaguliba = 2-3. 6. gulibaguliba = 3-3.

PORT ESSINGTON.[172]

1. erad. 2. nargarik. 3. nargarikelerad = 2-1. 4. nargariknargarik = 2-2.

WARREGO.

1. tarlina. 2. barkalo. 3. tarlina barkalo = 1-2.

CROCKER ISLAND.

1. roka. 2. orialk. 3. orialkeraroka = 2-1.

WARRIOR ISLAND.[173]

1. woorapoo. 2. ocasara. 3. ocasara woorapoo = 2-1. 4. ocasara ocasara = 2-2.

DIPPIL.[174]

1. kalim. 2. buller. 3. boppa. 4. buller gira buller = 2 + 2. 5. buller gira buller kalim = 2 + 2 + 1.

FRAZER'S ISLAND.[175]

1. kalim. 2. bulla. 3. goorbunda. 4. bulla-bulla = 2-2.

MORETON'S BAY.[176]

1. kunner. 2. budela. 3. muddan. 4. budela berdelu = 2-2.

ENCOUNTER BAY.[177]

1. yamalaitye. 2. ningenk. 3. nepaldar. 4. kuko kuko = 2-2, or pair pair. 5. kuko kuko ki = 2-2-1. 6. kuko kuko kuko = 2-2-2. 7. kuko kuko kuko ki = 2-2-2-1.

ADELAIDE.[178]

1. kuma. 2. purlaitye, or bula. 3. marnkutye. 4. yera-bula = pair 2. 5. yera-bula kuma = pair 2-1. 6. yera-bula purlaitye = pair 2.2.

WIRADUROI.[179]

1. numbai. 2. bula. 3. bula-numbai = 2-1. 4. bungu = many. 5. bungu-galan = very many.

WIRRI-WIRRI.[180]

1. mooray. 2. boollar. 3. belar mooray = 2-1. 4. boollar boollar = 2-2. 5. mongoonballa. 6. mongun mongun.

COOPER'S CREEK.[181]

1. goona. 2. barkoola. 3. barkoola goona = 2-1. 4. barkoola barkoola = 2-2.

BOURKE, DARLING RIVER.[182]

1. neecha. 2. boolla. 4. boolla neecha = 2-1. 3. boolla boolla = 2-2.

MURRAY RIVER, N.W. BEND.[183]

1. mata. 2. rankool. 3. rankool mata = 2-1. 4. rankool rankool = 2-2.

YIT-THA.[184]

1. mo. 2. thral. 3. thral mo = 2-1. 4. thral thral = 2-2.

PORT DARWIN.[185]

1. kulagook. 2. kalletillick. 3. kalletillick kulagook = 2-1. 4. kalletillick kalletillick = 2-2.

CHAMPION BAY.[186]

1. kootea. 2. woothera. 3. woothera kootea = 2-1. 4. woothera woothera = 2-2.

BELYANDO RIVER.[187]

1. wogin. 2. booleroo. 3. booleroo wogin = 2-1. 4. booleroo booleroo = 2-2.

WARREGO RIVER.

1. onkera. 2. paulludy. 3. paulludy onkera = 2-1. 4. paulludy paulludy = 2-2.

RICHMOND RIVER.

1. yabra. 2. booroora. 3. booroora yabra = 2-1. 4. booroora booroora = 2-2.

PORT MACQUARIE.

1. warcol. 2. blarvo. 3. blarvo warcol = 2-1. 4. blarvo blarvo = 2-2.

HILL END.

1. miko. 2. bullagut. 3. bullagut miko = 2-1. 4. bullagut bullagut = 2-2.

MONEROO 1. boor. 2. wajala, blala. 3. blala boor = 2-1. 4. wajala wajala.

GONN STATION.

1. karp. 2. pellige. 3. pellige karp = 2-1. 4. pellige pellige = 2-2.

UPPER YARRA.

1. kaambo. 2. benjero. 3. benjero kaambo = 2-2. 4. benjero on benjero = 2-2.

OMEO.

1. bore. 2. warkolala. 3. warkolala bore = 2-1. 4. warkolala warkolala = 2-2.

SNOWY RIVER.

1. kootook. 2. boolong. 3. booloom catha kootook = 2 + 1. 4. booloom catha booloom = 2 + 2.

NGARRIMOWRO. 1. warrangen. 2. platir. 3. platir warrangen = 2-1. 4. platir platir = 2-2.

This Australian list might be greatly extended, but the scales selected may be taken as representative examples of Australian binary scales. Nearly all of them show a structure too clearly marked to require comment. In a few cases, however, the systems are to be regarded rather as showing a trace of binary structure, than as perfect examples of counting by twos. Examples of this nature are especially numerous in Curr's extensive list—the most complete collection of Australian vocabularies ever made.

A few binary scales have been found in South America, but they show no important variation on the Australian systems cited above. The only ones I have been able to collect are the following:

BAKAIRI.[188]

1. tokalole. 2. asage. 3. asage tokalo = 2-1. 4. asage asage = 2-2.

ZAPARA.[189]

1. nuquaqui. 2. namisciniqui. 3. haimuckumarachi. 4. namisciniqui ckara maitacka = 2 + 2. 5. namisciniqui ckara maitacka nuquaqui = 2 pairs + 1. 6. haimuckumaracki ckaramsitacka = 3 pairs.

APINAGES.[190]

1. pouchi. 2. at croudou. 3. at croudi-pshi = 2-1. 4. agontad-acroudo = 2-2.

COTOXO.[191]

1. ihueto. 2. ize. 3. ize-te-hueto = 2-1. 4. ize-te-seze = 2-2. 5. ize-te-seze-hue = 2-2-1.

MBAYI.[192]

1. uninitegui. 2. iniguata. 3. iniguata dugani = 2 over. 4. iniguata driniguata = 2-2. 5. oguidi = many.

TAMA.[193]

1. teyo. 2. cayapa. 3. cho-teyo = 2 + 1. 4. cayapa-ria = 2 again. 5. cia-jente = hand.

CURETU.[194]

1. tchudyu. 2. ap-adyu. 3. arayu. 4. apaedyai = 2 + 2. 5. tchumupa.

If the existence of number systems like the above are to be accounted for simply on the ground of low civilization, one might reasonably expect to find ternary and and quaternary scales, as well as binary. Such scales actually exist, though not in such numbers as the binary. An example of the former is the Betoya scale,[195] which runs thus:

1. edoyoyoi. 2. edoi = another. 3. ibutu = beyond. 4. ibutu-edoyoyoi = beyond 1, or 3-1. 5. ru-mocoso = hand.

The Kamilaroi scale, given as an example of binary formation, is partly ternary; and its word for 6, guliba guliba, 3-3, is purely ternary. An occasional ternary trace is also found in number systems otherwise decimal or quinary vigesimal; as the dlkunoutl, second 3, of the Haida Indians of British Columbia. The Karens of India[196] in a system otherwise strictly decimal, exhibit the following binary-ternary-quaternary vagary:

6. then tho = 3 x 2. 7. then tho ta = 3 x 2-1. 8. lwie tho = 4 x 2. 9. lwie tho ta = 4 x 2-1.

In the Wokka dialect,[197] found on the Burnett River, Australia, a single ternary numeral is found, thus:

1. karboon. 2. wombura. 3. chrommunda. 4. chrommuda karboon = 3-1.

Instances of quaternary numeration are less rare than are those of ternary, and there is reason to believe that this method of counting has been practised more extensively than any other, except the binary and the three natural methods, the quinary, the decimal, and the vigesimal. The number of fingers on one hand is, excluding the thumb, four. Possibly there have been tribes among which counting by fours arose as a legitimate, though unusual, result of finger counting; just as there are, now and then, individuals who count on their fingers with the forefinger as a starting-point. But no such practice has ever been observed among savages, and such theorizing is the merest guess-work. Still a definite tendency to count by fours is sometimes met with, whatever be its origin. Quaternary traces are repeatedly to be found among the Indian languages of British Columbia. In describing the Columbians, Bancroft says: "Systems of numeration are simple, proceeding by fours, fives, or tens, according to the different languages...."[198] The same preference for four is said to have existed in primitive times in the languages of Central Asia, and that this form of numeration, resulting in scores of 16 and 64, was a development of finger counting.[199]

In the Hawaiian and a few other languages of the islands of the central Pacific, where in general the number systems employed are decimal, we find a most interesting case of the development, within number scales already well established, of both binary and quaternary systems. Their origin seems to have been perfectly natural, but the systems themselves must have been perfected very slowly. In Tahitian, Rarotongan, Mangarevan, and other dialects found in the neighbouring islands of those southern latitudes, certain of the higher units, tekau, rau, mano, which originally signified 10, 100, 1000, have become doubled in value, and now stand for 20, 200, 2000. In Hawaiian and other dialects they have again been doubled, and there they stand for 40, 400, 4000.[200] In the Marquesas group both forms are found, the former in the southern, the latter in the northern, part of the archipelago; and it seems probable that one or both of these methods of numeration are scattered somewhat widely throughout that region. The origin of these methods is probably to be found in the fact that, after the migration from the west toward the east, nearly all the objects the natives would ever count in any great numbers were small,—as yams, cocoanuts, fish, etc.,—and would be most conveniently counted by pairs. Hence the native, as he counted one pair, two pairs, etc., might readily say one, two, and so on, omitting the word "pair" altogether. Having much more frequent occasion to employ this secondary than the primary meaning of his numerals, the native would easily allow the original significations to fall into disuse, and in the lapse of time to be entirely forgotten. With a subsequent migration to the northward a second duplication might take place, and so produce the singular effect of giving to the same numeral word three different meanings in different parts of Oceania. To illustrate the former or binary method of numeration, the Tahuatan, one of the southern dialects of the Marquesas group, may be employed.[201] Here the ordinary numerals are:

1. tahi, 10. onohuu. 20. takau. 200. au. 2,000. mano. 20,000. tini. 20,000. tufa. 2,000,000. pohi.

In counting fish, and all kinds of fruit, except breadfruit, the scale begins with tauna, pair, and then, omitting onohuu, they employ the same words again, but in a modified sense. Takau becomes 10, au 100, etc.; but as the word "pair" is understood in each case, the value is the same as before. The table formed on this basis would be:

2 (units) = 1 tauna = 2. 10 tauna = 1 takau = 20. 10 takau = 1 au = 200. 10 au = 1 mano = 2000. 10 mano = 1 tini = 20,000. 10 tini = 1 tufa = 200,000. 10 tufa = 1 pohi = 2,000,000.

For counting breadfruit they use pona, knot, as their unit, breadfruit usually being tied up in knots of four. Takau now takes its third signification, 40, and becomes the base of their breadfruit system, so to speak. For some unknown reason the next unit, 400, is expressed by tauau, while au, which is the term that would regularly stand for that number, has, by a second duplication, come to signify 800. The next unit, mano, has in a similar manner been twisted out of its original sense, and in counting breadfruit is made to serve for 8000. In the northern, or Nukuhivan Islands, the decimal-quaternary system is more regular. It is in the counting of breadfruit only,[202]

4 breadfruits = 1 pona = 4. 10 pona = 1 toha = 40. 10 toha = 1 au = 400. 10 au = 1 mano = 4000. 10 mano = 1 tini = 40,000. 10 tini = 1 tufa = 400,000. 10 tufa = 1 pohi = 4,000,000.

In the Hawaiian dialect this scale is, with slight modification, the universal scale, used not only in counting breadfruit, but any other objects as well. The result is a complete decimal-quaternary system, such as is found nowhere else in the world except in this and a few of the neighbouring dialects of the Pacific. This scale, which is almost identical with the Nukuhivan, is[203]

4 units = 1 ha or tauna = 4. 10 tauna = 1 tanaha = 40. 10 tanaha = 1 lau = 400. 10 lau = 1 mano = 4000. 10 mano = 1 tini = 40,000. 10 tini = 1 lehu = 400,000.

The quaternary element thus introduced has modified the entire structure of the Hawaiian number system. Fifty is tanaha me ta umi, 40 + 10; 76 is 40 + 20 + 10 + 6; 100 is ua tanaha ma tekau, 2 x 40 + 10; 200 is lima tanaha, 5 x 40; and 864,895 is 2 x 400,000 + 40,000 + 6 x 4000 + 2 x 400 + 2 x 40 + 10 + 5.[204] Such examples show that this secondary influence, entering and incorporating itself as a part of a well-developed decimal system, has radically changed it by the establishment of 4 as the primary number base. The role which 10 now plays is peculiar. In the natural formation of a quaternary scale new units would be introduced at 16, 64, 256, etc.; that is, at the square, the cube, and each successive power of the base. But, instead of this, the new units are introduced at 10 x 4, 100 x 4, 1000 x 4, etc.; that is, at the products of 4 by each successive power of the old base. This leaves the scale a decimal scale still, even while it may justly be called quaternary; and produces one of the most singular and interesting instances of number-system formation that has ever been observed. In this connection it is worth noting that these Pacific island number scales have been developed to very high limits—in some cases into the millions. The numerals for these large numbers do not seem in any way indefinite, but rather to convey to the mind of the native an idea as clear as can well be conveyed by numbers of such magnitude. Beyond the limits given, the islanders have indefinite expressions, but as far as can be ascertained these are only used when the limits given above have actually been passed. To quote one more example, the Hervey Islanders, who have a binary-decimal scale, count as follows:

5 kaviri (bunches of cocoanuts) = 1 takau = 20. 10 takau = 1 rau = 200. 10 rau = 1 mano = 2000. 10 mano = 1 kiu = 20,000. 10 kiu = 1 tini = 200,000.

Anything above this they speak of in an uncertain way, as mano mano or tini tini, which may, perhaps, be paralleled by our English phrases "myriads upon myriads," and "millions of millions."[205] It is most remarkable that the same quarter of the globe should present us with the stunted number sense of the Australians, and, side by side with it, so extended and intelligent an appreciation of numerical values as that possessed by many of the lesser tribes of Polynesia.

The Luli of Paraguay[206] show a decided preference for the base 4. This preference gives way only when they reach the number 10, which is an ordinary digit numeral. All numbers above that point belong rather to decimal than to quaternary numeration. Their numerals are:

1. alapea. 2. tamop. 3. tamlip. 4. lokep. 5. lokep moile alapea = 4 with 1, or is-alapea = hand 1. 6. lokep moile tamop = 4 with 2. 7. lokep moile tamlip = 4 with 3. 8. lokep moile lokep = 4 with 4. 9. lokep moile lokep alapea = 4 with 4-1. 10. is yaoum = all the fingers of hand. 11. is yaoum moile alapea = all the fingers of hand with 1. 20. is elu yaoum = all the fingers of hand and foot. 30. is elu yaoum moile is-yaoum = all the fingers of hand and foot with all the fingers of hand.

Still another instance of quaternary counting, this time carrying with it a suggestion of binary influence, is furnished by the Mocobi[207] of the Parana region. Their scale is exceedingly rude, and they use the fingers and toes almost exclusively in counting; only using their spoken numerals when, for any reason, they wish to dispense with the aid of their hands and feet. Their first eight numerals are:

1. iniateda. 2. inabaca. 3. inabacao caini = 2 above. 4. inabacao cainiba = 2 above 2; or natolatata. 5. inibacao cainiba iniateda = 2 above 2-1; or natolatata iniateda = 4-1. 6. natolatatata inibaca = 4-2. 7. natolata inibacao-caini = 4-2 above. 8. natolata-natolata = 4-4.

There is probably no recorded instance of a number system formed on 6, 7, 8, or 9 as a base. No natural reason exists for the choice of any of these numbers for such a purpose; and it is hardly conceivable that any race should proceed beyond the unintelligent binary or quaternary stage, and then begin the formation of a scale for counting with any other base than one of the three natural bases to which allusion has already been made. Now and then some anomalous fragment is found imbedded in an otherwise regular system, which carries us back to the time when the savage was groping his way onward in his attempt to give expression to some number greater than any he had ever used before; and now and then one of these fragments is such as to lead us to the border land of the might-have-been, and to cause us to speculate on the possibility of so great a numerical curiosity as a senary or a septenary scale. The Bretons call 18 triouec'h, 3-6, but otherwise their language contains no hint of counting by sixes; and we are left at perfect liberty to theorize at will on the existence of so unusual a number word. Pott remarks[208] that the Bolans, of western Africa, appear to make some use of 6 as their number base, but their system, taken as a whole, is really a quinary-decimal. The language of the Sundas,[209] or mountaineers of Java, contains traces of senary counting. The Akra words for 7 and 8, paggu and paniu, appear to mean 6-1 and 7-1, respectively; and the same is true of the corresponding Tambi words pagu and panjo.[210] The Watji tribe[211] call 6 andee, and 7 anderee, which probably means 6-1. These words are to be regarded as accidental variations on the ordinary laws of formation, and are no more significant of a desire to count by sixes than is the Wallachian term deu-maw, which expresses 18 as 2-9, indicates the existence of a scale of which 9 is the base. One remarkably interesting number system is that exhibited by the Mosquito tribe[212] of Central America, who possess an extensive quinary-vigesimal scale containing one binary and three senary compounds. The first ten words of this singular scale, which has already been quoted, are:

1. kumi. 2. wal. 3. niupa. 4. wal-wal = 2-2. 5. mata-sip = fingers of one hand. 6. matlalkabe. 7. matlalkabe pura kumi = 6 + 1. 8. matlalkabe pura wal = 6 + 2. 9. matlalkabe pura niupa = 6 + 3. 10. mata-wal-sip = fingers of the second hand.

In passing from 6 to 7, this tribe, also, has varied the almost universal law of progression, and has called 7 6-1. Their 8 and 9 are formed in a similar manner; but at 10 the ordinary method is resumed, and is continued from that point onward. Few number systems contain as many as three numerals which are associated with 6 as their base. In nearly all instances we find such numerals singly, or at most in pairs; and in the structure of any system as a whole, they are of no importance whatever. For example, in the Pawnee, a pure decimal scale, we find the following odd sequence:[213]

6. shekshabish. 7. petkoshekshabish = 2-6, i.e. 2d 6. 8. touwetshabish = 3-6, i.e. 3d 6. 9. loksherewa = 10 - 1.

In the Uainuma scale the expressions for 7 and 8 are obviously referred to 6, though the meaning of 7 is not given, and it is impossible to guess what it really does signify. The numerals in question are:[214]

6. aira-ettagapi. 7. aira-ettagapi-hairiwigani-apecapecapsi. 8. aira-ettagapi-matschahma = 6 + 2.

In the dialect of the Mille tribe a single trace of senary counting appears, as the numerals given below show:[215]

6. dildjidji. 7. dildjidji me djuun = 6 + 1.

Finally, in the numerals used by the natives of the Marshall Islands, the following curiously irregular sequence also contains a single senary numeral:[216]

6. thil thino = 3 + 3. 7. thilthilim-thuon = 6 + 1. 8. rua-li-dok = 10 - 2. 9. ruathim-thuon = 10 - 2 + 1.

Many years ago a statement appeared which at once attracted attention and awakened curiosity. It was to the effect that the Maoris, the aboriginal inhabitants of New Zealand, used as the basis of their numeral system the number 11; and that the system was quite extensively developed, having simple words for 121 and 1331, i.e. for the square and cube of 11. No apparent reason existed for this anomaly, and the Maori scale was for a long time looked upon as something quite exceptional and outside all ordinary rules of number-system formation. But a closer and more accurate knowledge of the Maori language and customs served to correct the mistake, and to show that this system was a simple decimal system, and that the error arose from the following habit. Sometimes when counting a number of objects the Maoris would put aside 1 to represent each 10, and then those so set aside would afterward be counted to ascertain the number of tens in the heap. Early observers among this people, seeing them count 10 and then set aside 1, at the same time pronouncing the word tekau, imagined that this word meant 11, and that the ignorant savage was making use of this number as his base. This misconception found its way into the early New Zealand dictionary, but was corrected in later editions. It is here mentioned only because of the wide diffusion of the error, and the interest it has always excited.[217]

Aside from our common decimal scale, there exist in the English language other methods of counting, some of them formal enough to be dignified by the term system—as the sexagesimal method of measuring time and angular magnitude; and the duodecimal system of reckoning, so extensively used in buying and selling. Of these systems, other than decimal, two are noticed by Tylor,[218] and commented on at some length, as follows:

"One is the well-known dicing set, ace, deuce, tray, cater, cinque, size; thus size-ace is 6-1, cinques or sinks, double 5. These came to us from France, and correspond with the common French numerals, except ace, which is Latin as, a word of great philological interest, meaning 'one.' The other borrowed set is to be found in the Slang Dictionary. It appears that the English street-folk have adopted as a means of secret communication a set of Italian numerals from the organ-grinders and image-sellers, or by other ways through which Italian or Lingua Franca is brought into the low neighbourhoods of London. In so doing they have performed a philological operation not only curious but instructive. By copying such expressions as due soldi, tre soldi, as equivalent to 'twopence,' 'threepence,' the word saltee became a recognized slang term for 'penny'; and pence are reckoned as follows:

oney saltee 1d. uno soldo. dooe saltee 2d. due soldi. tray saltee 3d. tre soldi. quarterer saltee 4d. quattro soldi. chinker saltee 5d. cinque soldi. say saltee 6d. sei soldi. say oney saltee, or setter saltee 7d. sette soldi. say dooe saltee, or otter saltee 8d. otto soldi. say tray saltee, or nobba saltee 9d. nove soldi. say quarterer saltee, or dacha saltee 10d. dieci soldi. say chinker saltee or dacha oney saltee 11d. undici soldi. oney beong 1s. a beong say saltee 1s. 6d. dooe beong say saltee, or madza caroon 2s. 6d. (half-crown, mezza corona).

One of these series simply adopts Italian numerals decimally. But the other, when it has reached 6, having had enough of novelty, makes 7 by 6-1, and so forth. It is for no abstract reason that 6 is thus made the turning-point, but simply because the costermonger is adding pence up to the silver sixpence, and then adding pence again up to the shilling. Thus our duodecimal coinage has led to the practice of counting by sixes, and produced a philological curiosity, a real senary notation."

In addition to the two methods of counting here alluded to, another may be mentioned, which is equally instructive as showing how readily any special method of reckoning may be developed out of the needs arising in connection with any special line of work. As is well known, it is the custom in ocean, lake, and river navigation to measure soundings by the fathom. On the Mississippi River, where constant vigilance is needed because of the rapid shifting of sand-bars, a special sounding nomenclature has come into vogue,[219] which the following terms will illustrate:

5 ft. = five feet. 6 ft. = six feet. 9 ft. = nine feet. 10-1/2 ft. = a quarter less twain; i.e. a quarter of a fathom less than 2. 12 ft. = mark twain. 13-1/2 ft. = a quarter twain. 16-1/2 ft. = a quarter less three. 18 ft. = mark three. 19-1/2 ft. = a quarter three. 24 ft. = deep four.

As the soundings are taken, the readings are called off in the manner indicated in the table; 10-1/2 feet being "a quarter less twain," 12 feet "mark twain," etc. Any sounding above "deep four" is reported as "no bottom." In the Atlantic and Gulf waters on the coast of this country the same system prevails, only it is extended to meet the requirements of the deeper soundings there found, and instead of "six feet," "mark twain," etc., we find the fuller expressions, "by the mark one," "by the mark two," and so on, as far as the depth requires. This example also suggests the older and far more widely diffused method of reckoning time at sea by bells; a system in which "one bell," "two bells," "three bells," etc., mark the passage of time for the sailor as distinctly as the hands of the clock could do it. Other examples of a similar nature will readily suggest themselves to the mind.

Two possible number systems that have, for purely theoretical reasons, attracted much attention, are the octonary and the duodecimal systems. In favour of the octonary system it is urged that 8 is an exact power of 2; or in other words, a large number of repeated halves can be taken with 8 as a starting-point, without producing a fractional result. With 8 as a base we should obtain by successive halvings, 4, 2, 1. A similar process in our decimal scale gives 5, 2-1/2, 1-1/4. All this is undeniably true, but, granting the argument up to this point, one is then tempted to ask "What of it?" A certain degree of simplicity would thereby be introduced into the Theory of Numbers; but the only persons sufficiently interested in this branch of mathematics to appreciate the benefit thus obtained are already trained mathematicians, who are concerned rather with the pure science involved, than with reckoning on any special base. A slightly increased simplicity would appear in the work of stockbrokers, and others who reckon extensively by quarters, eighths, and sixteenths. But such men experience no difficulty whatever in performing their mental computations in the decimal system; and they acquire through constant practice such quickness and accuracy of calculation, that it is difficult to see how octonary reckoning would materially assist them. Altogether, the reasons that have in the past been adduced in favour of this form of arithmetic seem trivial. There is no record of any tribe that ever counted by eights, nor is there the slightest likelihood that such a system could ever meet with any general favour. It is said that the ancient Saxons used the octonary system,[220] but how, or for what purposes, is not stated. It is not to be supposed that this was the common system of counting, for it is well known that the decimal scale was in use as far back as the evidence of language will take us. But the field of speculation into which one is led by the octonary scale has proved most attractive to some, and the conclusion has been soberly reached, that in the history of the Aryan race the octonary was to be regarded as the predecessor of the decimal scale. In support of this theory no direct evidence is brought forward, but certain verbal resemblances. Those ignes fatuii of the philologist are made to perform the duty of supporting an hypothesis which would never have existed but for their own treacherous suggestions. Here is one of the most attractive of them:

Between the Latin words novus, new, and novem, nine, there exists a resemblance so close that it may well be more than accidental. Nine is, then, the new number; that is, the first number on a new count, of which 8 must originally have been the base. Pursuing this thought by investigation into different languages, the same resemblance is found there. Hence the theory is strengthened by corroborative evidence. In language after language the same resemblance is found, until it seems impossible to doubt, that in prehistoric times, 9 was the new number—the beginning of a second tale. The following table will show how widely spread is this coincidence:

Sanskrit, navan = 9. nava = new. Persian, nuh = 9. nau = new. Greek, [Greek: ennea] = 9. [Greek: neos] = new. Latin, novem = 9. novus = new. German, neun = 9. neu = new. Swedish, nio = 9. ny = new. Dutch, negen = 9. nieuw = new. Danish, ni = 9. ny = new. Icelandic, nyr = 9. niu = new. English, nine = 9. new = new. French, neuf = 9. nouveau = new. Spanish, nueve = 9. neuvo = new. Italian, nove = 9. nuovo = new. Portuguese, nove = 9. novo = new. Irish, naoi = 9. nus = new. Welsh, naw = 9. newydd = new. Breton, nevez = 9. nuhue = new.[221]

This table might be extended still further, but the above examples show how widely diffused throughout the Aryan languages is this resemblance. The list certainly is an impressive one, and the student is at first thought tempted to ask whether all these resemblances can possibly have been accidental. But a single consideration sweeps away the entire argument as though it were a cobweb. All the languages through which this verbal likeness runs are derived directly or indirectly from one common stock; and the common every-day words, "nine" and "new," have been transmitted from that primitive tongue into all these linguistic offspring with but little change. Not only are the two words in question akin in each individual language, but they are akin in all the languages. Hence all these resemblances reduce to a single resemblance, or perhaps identity, that between the Aryan words for "nine" and "new." This was probably an accidental resemblance, no more significant than any one of the scores of other similar cases occurring in every language. If there were any further evidence of the former existence of an Aryan octonary scale, the coincidence would possess a certain degree of significance; but not a shred has ever been produced which is worthy of consideration. If our remote ancestors ever counted by eights, we are entirely ignorant of the fact, and must remain so until much more is known of their language than scholars now have at their command. The word resemblances noted above are hardly more significant than those occurring in two Polynesian languages, the Fatuhivan and the Nakuhivan,[222] where "new" is associated with the number 7. In the former case 7 is fitu, and "new" is fou; in the latter 7 is hitu, and "new" is hou. But no one has, because of this likeness, ever suggested that these tribes ever counted by the senary method. Another equally trivial resemblance occurs in the Tawgy and the Kamassin languages,[223] thus:

TAWGY. KAMASSIN.

8. siti-data = 2 x 4. 8. sin-the'de = 2 x 4. 9. nameaitjuma = another. 9. amithun = another.

But it would be childish to argue, from this fact alone, that either 4 or 8 was the number base used.

In a recent antiquarian work of considerable interest, the author examines into the question of a former octonary system of counting among the various races of the world, particularly those of Asia, and brings to light much curious and entertaining material respecting the use of this number. Its use and importance in China, India, and central Asia, as well as among some of the islands of the Pacific, and in Central America, leads him to the conclusion that there was a time, long before the beginning of recorded history, when 8 was the common number base of the world. But his conclusion has no basis in his own material even. The argument cannot be examined here, but any one who cares to investigate it can find there an excellent illustration of the fact that a pet theory may take complete possession of its originator, and reduce him finally to a state of infantile subjugation.[224]

Of all numbers upon which a system could be based, 12 seems to combine in itself the greatest number of advantages. It is capable of division by 2, 3, 4, and 6, and hence admits of the taking of halves, thirds, quarters, and sixths of itself without the introduction of fractions in the result. From a commercial stand-point this advantage is very great; so great that many have seriously advocated the entire abolition of the decimal scale, and the substitution of the duodecimal in its stead. It is said that Charles XII. of Sweden was actually contemplating such a change in his dominions at the time of his death. In pursuance of this idea, some writers have gone so far as to suggest symbols for 10 and 11, and to recast our entire numeral nomenclature to conform to the duodecimal base.[225] Were such a change made, we should express the first nine numbers as at present, 10 and 11 by new, single symbols, and 12 by 10. From this point the progression would be regular, as in the decimal scale—only the same combination of figures in the different scales would mean very different things. Thus, 17 in the decimal scale would become 15 in the duodecimal; 144 in the decimal would become 100 in the duodecimal; and 1728, the cube of the new base, would of course be represented by the figures 1000.

It is impossible that any such change can ever meet with general or even partial favour, so firmly has the decimal scale become intrenched in its position. But it is more than probable that a large part of the world of trade and commerce will continue to buy and sell by the dozen, the gross, or some multiple or fraction of the one or the other, as long as buying and selling shall continue. Such has been its custom for centuries, and such will doubtless be its custom for centuries to come. The duodecimal is not a natural scale in the same sense as are the quinary, the decimal, and the vigesimal; but it is a system which is called into being long after the complete development of one of the natural systems, solely because of the simple and familiar fractions into which its base is divided. It is the scale of civilization, just as the three common scales are the scales of nature. But an example of its use was long sought for in vain among the primitive races of the world. Humboldt, in commenting on the number systems of the various peoples he had visited during his travels, remarked that no race had ever used exclusively that best of bases, 12. But it has recently been announced[226] that the discovery of such a tribe had actually been made, and that the Aphos of Benue, an African tribe, count to 12 by simple words, and then for 13 say 12-1, for 14, 12-2, etc. This report has yet to be verified, but if true it will constitute a most interesting addition to anthropological knowledge.