The Number Concept: Its Origin and Development

More rarely yet are instances met with of languages which make use of subtraction almost as freely as addition, in the composition of numerals. Within the past few years such an instance has been noticed in the case of the Bellacoola language of British Columbia. In their numeral scale 15, “one foot,” is followed by 16, “one man less 4”; 17, “one man less 3”; 18, “one man less 2”; 19, “one man less 1”; and 20, one man. Twenty-five is “one man and one hand”; 26, “one man and two hands less 4”; 36, “two men less 4”; and so on. This method of formation prevails throughout the entire numeral scale.⁶³

One of the best known and most interesting examples of subtraction as a well-defined principle of formation is found in the Maya scale. Up to 40 no special peculiarity appears; but as the count progresses beyond that point we find a succession of numerals which one is almost tempted to call 60 − 19, 60 − 18, 60 − 17, etc. Literally translated the meanings seem to be 1 to 60, 2 to 60, 3 to 60, etc. The point of reference is 60, and the thought underlying the words may probably be expressed by the paraphrases, “1 on the third score, 2 on the third score, 3 on the third score,” etc. Similarly, 61 is 1 on the fourth score, 81 is one on the fifth score, 381 is 1 on the nineteenth score, and so on to 400. At 441 the same formation reappears; and it continues to characterize the system in a regular and consistent manner, no matter how far it is extended.⁶⁴

The Yoruba language of Africa is another example of most lavish use of subtraction; but it here results in a system much less consistent and natural than that just considered. Here we find not only 5, 10, and 20 subtracted from the next higher unit, but also 40, and even 100. For example, 360 is 400 − 40; 460 is 500 − 40; 500 is 600 − 100; 1300 is 1400 − 100, etc. One of the Yoruba units is 200; and all the odd hundreds up to 2000, the next higher unit, are formed by subtracting 100 from the next higher multiple of 200. The system is quite complex, and very artificial; and seems to have been developed by intercourse with traders.⁶⁵

It has already been stated that the primitive meanings of our own simple numerals have been lost. This is also true of the languages of nearly all other civilized peoples, and of numerous savage races as well. We are at liberty to suppose, and we do suppose, that in very many cases these words once expressed meanings closely connected with the names of the fingers, or with the fingers themselves, or both. Now and then a case is met with in which the numeral word frankly avows its meaning—as in the Botocudo language, where 1 is expressed by podzik, finger, and 2 by kripo, double finger;⁶⁶ and in the Eskimo dialect of Hudson's Bay, where eerkitkoka means both 10 and little finger.⁶⁷ Such cases are, however, somewhat exceptional.

In a few noteworthy instances, the words composing the numeral scale of a language have been carefully investigated and their original meanings accurately determined. The simple structure of many of the rude languages of the world should render this possible in a multitude of cases; but investigators are too often content with the mere numerals themselves, and make no inquiry respecting their meanings. But the following exposition of the Zuñi scale, given by Lieutenant Gushing⁶⁸ leaves nothing to be desired:

This finishes the list of original simple numerals, the Zuñi stopping, or “notching off,” when he finishes the fingers of one hand. Compounding now begins.

The process of formation indicated in 11 is used in the succeeding numerals up to 19.

The only numerals calling for any special note are those for 11 and 9. For 9 we should naturally expect a word corresponding in structure and meaning to the words for 7 and 8. But instead of the “four brought to and held up with the rest,” for which we naturally look, the Zuñi, to show that he has used all of his fingers but one, says “all but all are held up with the rest.” To express 11 he cannot use a similar form of composition, since he has already used it in constructing his word for 6, so he says “all the fingers and another over above held.”

The one remarkable point to be noted about the Zuñi scale is, after all, the formation of the words for 1 and 2. While the savage almost always counts on his fingers, it does not seem at all certain that these words would necessarily be of finger formation. The savage can always distinguish between one object and two objects, and it is hardly reasonable to believe that any external aid is needed to arrive at a distinct perception of this difference. The numerals for 1 and 2 would be the earliest to be formed in any language, and in most, if not all, cases they would be formed long before the need would be felt for terms to describe any higher number. If this theory be correct, we should expect to find finger names for numerals beginning not lower than 3, and oftener with 5 than with any other number. The highest authority has ventured the assertion that all numeral words have their origin in the names of the fingers;⁶⁹ substantially the same conclusion was reached by Professor Pott, of Halle, whose work on numeral nomenclature led him deeply into the study of the origin of these words. But we have abundant evidence at hand to show that, universal as finger counting has been, finger origin for numeral words has by no means been universal. That it is more frequently met with than any other origin is unquestionably true; but in many instances, which will be more fully considered in the following chapter, we find strictly non-digital derivations, especially in the case of the lowest members of the scale. But in nearly all languages the origin of the words for 1, 2, 3, and 4 are so entirely unknown that speculation respecting them is almost useless.

An excellent illustration of the ordinary method of formation which obtains among number scales is furnished by the Eskimos of Point Barrow,⁷⁰ who have pure numeral words up to 5, and then begin a systematic course of word formation from the names of their fingers. If the names of the first five numerals are of finger origin, they have so completely lost their original form, or else the names of the fingers themselves have so changed, that no resemblance is now to be detected between them. This scale is so interesting that it is given with considerable fulness, as follows:

In this scale we find the finger origin appearing so clearly and so repeatedly that one feels some degree of surprise at finding 5 expressed by a pure numeral instead of by some word meaning hand or fingers of one hand. In this respect the Eskimo dialects are somewhat exceptional among scales built up of digital words. The system of the Greenland Eskimos, though differing slightly from that of their Point Barrow cousins, shows the same peculiarity. The first ten numerals of this scale are:⁷¹

The same process is now repeated, only the feet instead of the hands are used; and the completion of the second 10 is marked by the word innuk, man. It may be that the Eskimo word for 5 is, originally, a digital word, but if so, the fact has not yet been detected. From the analogy furnished by other languages we are justified in suspecting that this may be the case; for whenever a number system contains digital words, we expect them to begin with five, as, for example, in the Arawak scale,⁷² which runs:

The four sets of numerals just given may be regarded as typifying one of the most common forms of primitive counting; and the words they contain serve as illustrations of the means which go to make up the number scales of savage races. Frequently the finger and toe origin of numerals is perfectly apparent, as in the Arawak system just given, which exhibits the simplest and clearest possible method of formation. Another even more interesting system is that of the Montagnais of northern Canada.⁷³ Here, as in the Zuñi scale, the words are digital from the outset.

The formation of 6, 7, and 8 of this scale is somewhat different from that ordinarily found. To express 6, the Montagnais separates the thumb and forefinger from the three remaining fingers of the left hand, and bringing the thumb of the right hand close to them, says: “3 from each side.” For 7 he either subtracts from 10, saying: “there are still 3 of them,” or he brings the thumb and forefinger of the right hand up to the thumb of the left, and says: “on one side there are 4 of them.” He calls 8 by the same name as many of the other Canadian tribes, that is, two 4's; and to show the proper number of fingers, he closes the thumb and little finger of the right hand, and then puts the three remaining fingers beside the thumb of the left hand. This method is, in some of these particulars, different from any other I have ever examined.

It often happens that the composition of numeral words is less easily understood, and the original meanings more difficult to recover, than in the examples already given. But in searching for number systems which show in the formation of their words the influence of finger counting, it is not unusual to find those in which the derivation from native words signifying finger, hand, toe, foot, and man, is just as frankly obvious as in the case of the Zuñi, the Arawak, the Eskimo, or the Montagnais scale. Among the Tamanacs,⁷⁴ one of the numerous Indian tribes of the Orinoco, the numerals are as strictly digital as in any of the systems already examined. The general structure of the Tamanac scale is shown by the following numerals:

In the Guarani⁷⁵ language of Paraguay the same method is found, with a different form of expression for 20. Here the numerals in question are

Another slight variation is furnished by the Kiriri language,⁷⁶ which is also one of the numerous South American Indian forms of speech, where we find the words to be

Illustrations of this kind might be multiplied almost indefinitely; and it is well to note that they may be drawn from all parts of the world. South America is peculiarly rich in native numeral words of this kind; and, as the examples above cited show, it is the field to which one instinctively turns when this subject is under discussion. The Zamuco numerals are, among others, exceedingly interesting, giving us still a new variation in method. They are⁷⁷

As is here indicated, the form of progression from 5 to 10, which we should expect to be “hand-1,” or “hand-and-1,” or some kindred expression, signifying that one hand had been completed, is simply “1 on the other.” Again, the expressions for 11, 12, etc., are merely “1 on the foot,” “2 on the foot,” etc., while 20 is “both feet ended.”

An equally interesting scale is furnished by the language of the Maipures⁷⁸ of the Orinoco, who count

In all the examples thus far given, 20 is expressed either by the equivalent of “man” or by some formula introducing the word “feet.” Both these modes of expressing what our own ancestors termed a “score,” are so common that one hesitates to say which is of the more frequent use. The following scale, from one of the Betoya dialects⁷⁹ of South America, is quite remarkable among digital scales, making no use of either “man” or “foot,” but reckoning solely by fives, or hands, as the numerals indicate.

In the last chapter mention was made of the scanty numeral systems of the Australian tribes, but a single scale was alluded to as reaching the comparatively high limit of 20. This system is that belonging to the Pikumbuls,⁸⁰ and the count runs thus:

As has already been stated, there is good ground for believing that this system was originally as limited as those obtained from other Australian tribes, and that its extension from 4, or perhaps from 5 onward, is of comparatively recent date.

A somewhat peculiar numeral nomenclature is found in the language of the Klamath Indians of Oregon. The first ten words in the Klamath scale are:⁸¹

In describing this system Mr. Gatschet says: “If the origin of the Klamath numerals is thus correctly traced, their inventors must have counted only the four long fingers without the thumb, and 5 was counted while saying hand away! hand off! The ‘four,’ or hand high! hand up! intimates that the hand was held up high after counting its four digits; and some term expressing this gesture was, in the case of nine, substituted by ‘one left over’ … which means to say, ‘only one is left until all the fingers are counted.’” It will be observed that the Klamath introduces not only the ordinary finger manipulation, but a gesture of the entire hand as well. It is a common thing to find something of the kind to indicate the completion of 5 or 10, and in one or two instances it has already been alluded to. Sometimes one or both of the closed fists are held up; sometimes the open hand, with all the fingers extended, is used; and sometimes an entirely independent gesture is introduced. These are, in general, of no special importance; but one custom in vogue among some of the prairie tribes of Indians, to which my attention was called by Dr. J. Owen Dorsey,⁸² should be mentioned. It is a gesture which signifies multiplication, and is performed by throwing the hand to the left. Thus, after counting 5, a wave of the hand to the left means 50. As multiplication is rather unusual among savage tribes, this is noteworthy, and would seem to indicate on the part of the Indian a higher degree of intelligence than is ordinarily possessed by uncivilized races.

In the numeral scale as we possess it in English, we find it necessary to retain the name of the last unit of each kind used, in order to describe definitely any numeral employed. Thus, fifteen, one hundred forty-two, six thousand seven hundred twenty-seven, give in full detail the numbers they are intended to describe. In primitive scales this is not always considered necessary; thus, the Zamucos express their teens without using their word for 10 at all. They say simply, 1 on the foot, 2 on the foot, etc. Corresponding abbreviations are often met; so often, indeed, that no further mention of them is needed. They mark one extreme, the extreme of brevity, found in the savage method of building up hand, foot, and finger names for numerals; while the Zuñi scale marks the extreme of prolixity in the formation of such words. A somewhat ruder composition than any yet noticed is shown in the numerals of the Vilelo scale,⁸³ which are:

In the examples thus far given, it will be noticed that the actual names of individual fingers do not appear. In general, such words as thumb, forefinger, little finger, are not found, but rather the hand-1, 1 on the next, or 1 over and above, which we have already seen, are the type forms for which we are to look. Individual finger names do occur, however, as in the scale of the Hudson's Bay Eskimos,⁸⁴ where the three following words are used both as numerals and as finger names:

Words of similar origin are found in the original Jiviro scale,⁸⁵ where the native numerals are:

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. Trumbull⁸⁶ 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

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 Africa⁹⁰ 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 is⁹²

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:

In the Choctaw language⁹⁵ the relation between 2 and 7, and 3 and 8, is no less clear. Here the words are:

The Nez Percés⁹⁶ repeat the first three words of their scale in their 6, 7, and 8 respectively, as a comparison of these numerals will show.

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

A precisely similar method of numeral building is used by some of our Western Indian tribes. Selecting a few of the Assiniboine numerals⁹⁸ as an illustration, we have

A still more primitive structure is shown in the numerals of the Mboushas⁹⁹ 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:

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.

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:

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 Africa¹⁰³ and that of the Mosquitos¹⁰⁴ 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.

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 Yoruba¹⁰⁵ 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:

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:

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.

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 thanthat 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-Mbutti¹¹¹ 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

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 are¹¹⁹

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 Greenlanders¹²⁰ 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 Mabas¹²¹ call 10 atuk, great 1; the Hottentots¹²² and the Hidatsa Indians call 100 great 10, their words being gei disi and pitikitstia respectively.

The Nicaraguans¹²³ 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 Dakota¹²⁴ 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”: