The Number Concept: Its Origin and Development

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 Zuñi, 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 Leone¹²⁶ 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 is¹²⁷ roambistelo polo amby valonjato amby arivo, 2 + 30 + 800 + 1000. The Aztec equivalent for 399 is¹²⁸ caxtolli onnauh poalli ipan caxtolli onnaui, (15 + 4) × 20 + 15 + 4; and the Sioux require for 29 the ponderous combination¹²⁹ 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 Maré 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 × 1000 × 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 Labillardière 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 Déné 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:

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.”

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

The same formation appears in the numerals of the ancient Phœnicians,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

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 respectively¹⁵³—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:

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 dialects¹⁶⁰ 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 × 5)(2 × 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 Nusqually¹⁶² of Puget Sound for 1000, i.e. paduts-subquätche, 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.

Chapter V.

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

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

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.

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:

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 India¹⁹⁶ in a system otherwise strictly decimal, exhibit the following binary-ternary-quaternary vagary:

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

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:

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:

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

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, is²⁰³

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 × 40 + 10; 200 is lima tanaha, 5 × 40; and 864,895 is 2 × 400,000 + 40,000 + 6 × 4000 + 2 × 400 + 2 × 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 × 4, 100 × 4, 1000 × 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:

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 Paraguay²⁰⁶ 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: