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diff --git a/16449.txt b/16449.txt new file mode 100644 index 0000000..a443ba8 --- /dev/null +++ b/16449.txt @@ -0,0 +1,8973 @@ +The Project Gutenberg EBook of The Number Concept, by Levi Leonard Conant + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: The Number Concept + Its Origin and Development + +Author: Levi Leonard Conant + +Release Date: August 5, 2005 [EBook #16449] + +Language: English + +Character set encoding: ASCII + +*** START OF THIS PROJECT GUTENBERG EBOOK THE NUMBER CONCEPT *** + + + + +Produced by Jonathan Ingram, Hagen von Eitzen and the +Online Distributed Proofreading Team at https://www.pgdp.net + + + + + +[*Transcriber's Note: +The following errors found in the original have been left as is. +Chapter I, 14th paragraph: + drop double quote before 'It is said'; +Chapter IV, 1st paragraph: + 'so similar than' read 'so similar that'; +Chapter IV, table of Hebrew numerals (near footnote 144): + insert comma after 'shemoneh'; +Chapter V, table of Tahuatan numerals (near footnote 201): + 'tahi,' read 'tahi.'; +Same table: + ' 20,000. tufa' read '200,000. tufa'; +Chapter VI, table of Bagrimma numerals (near footnote 259): + 'marta = 5 + 2' read 'marta = 5 + 3'; +Same table: + 'do-so = [5] + 3' read 'do-so = [5] + 4'; +Chapter VII, table of Nahuatl numerals (near footnote 365): + '90-10' read '80-10'; +In paragraph following that table: + '+ (15 + 4) x 400 x 800' read + '(15 + 4) x 20 x 400 x 8000 + (15 + 4) x 400 x 8000'; +In text of footnote 297: + 'II. I. p. 179' read 'II. i. p. 179'; +*] + + +THE MACMILLAN COMPANY +NEW YORK . BOSTON . CHICAGO . DALLAS +ATLANTA . SAN FRANCISCO + + + +MACMILLAN & CO., LIMITED +LONDON . BOMBAY . CALCUTTA +MELBOURNE + + + +THE MACMILLAN COMPANY +OF CANADA, LIMITED +TORONTO + + + + + + +THE NUMBER CONCEPT + + + +ITS ORIGIN AND DEVELOPMENT + + + +BY +LEVI LEONARD CONANT, PH.D. +ASSOCIATE PROFESSOR OF MATHEMATICS IN THE WORCESTER +POLYTECHNIC INSTITUTE + + + +New York +MACMILLAN AND CO. +AND LONDON +1931 + + + +COPYRIGHT, 1896, +BY THE MACMILLAN COMPANY. + + + + + + +COPYRIGHT, 1924, +BY EMMA B. CONANT. + + + + + + +All rights reserved--no part of this book may be reproduced in any form +without permission in writing from the publisher. + + + + + + +Set up and electrotyped. Published July, 1896. + + + +Norwood Press +J.S. Cushing Co.--Berwick & Smith Co. +Norwood, Mass., U.S.A. + + + + + +PREFACE. + + +In the selection of authorities which have been consulted in the +preparation of this work, and to which reference is made in the following +pages, great care has been taken. Original sources have been drawn upon in +the majority of cases, and nearly all of these are the most recent +attainable. Whenever it has not been possible to cite original and recent +works, the author has quoted only such as are most standard and +trustworthy. In the choice of orthography of proper names and numeral +words, the forms have, in almost all cases, been written as they were +found, with no attempt to reduce them to a systematic English basis. In +many instances this would have been quite impossible; and, even if +possible, it would have been altogether unimportant. Hence the forms, +whether German, French, Italian, Spanish, or Danish in their transcription, +are left unchanged. Diacritical marks are omitted, however, since the +proper key could hardly be furnished in a work of this kind. + +With the above exceptions, this study will, it is hoped, be found to be +quite complete; and as the subject here investigated has never before been +treated in any thorough and comprehensive manner, it is hoped that this +book may be found helpful. The collections of numeral systems illustrating +the use of the binary, the quinary, and other number systems, are, taken +together, believed to be the most extensive now existing in any language. +Only the cardinal numerals have been considered. The ordinals present no +marked peculiarities which would, in a work of this kind, render a separate +discussion necessary. Accordingly they have, though with some reluctance, +been omitted entirely. + +Sincere thanks are due to those who have assisted the author in the +preparation of his materials. Especial acknowledgment should be made to +Horatio Hale, Dr. D.G. Brinton, Frank Hamilton Cushing, and Dr. A.F. +Chamberlain. + +WORCESTER, MASS., Nov. 12, 1895. + + + + + +CONTENTS. + + +Chapter I. +Counting 1 +Chapter II. +Number System Limits 21 +Chapter III. +Origin of Number Words 37 +Chapter IV. +Origin of Number Words (_continued_) 74 +Chapter V. +Miscellaneous Number Bases 100 +Chapter VI. +The Quinary System 134 +Chapter VII. +The Vigesimal System 176 + * * * * * +Index 211 + + + + + + +THE NUMBER CONCEPT: ITS ORIGIN AND DEVELOPMENT. + + + + + + + +CHAPTER I. + +COUNTING. + + +Among the speculative questions which arise in connection with the study of +arithmetic from a historical standpoint, the origin of number is one that +has provoked much lively discussion, and has led to a great amount of +learned research among the primitive and savage languages of the human +race. A few simple considerations will, however, show that such research +must necessarily leave this question entirely unsettled, and will indicate +clearly that it is, from the very nature of things, a question to which no +definite and final answer can be given. + +Among the barbarous tribes whose languages have been studied, even in a +most cursory manner, none have ever been discovered which did not show some +familiarity with the number concept. The knowledge thus indicated has often +proved to be most limited; not extending beyond the numbers 1 and 2, or 1, +2, and 3. Examples of this poverty of number knowledge are found among the +forest tribes of Brazil, the native races of Australia and elsewhere, and +they are considered in some detail in the next chapter. At first thought it +seems quite inconceivable that any human being should be destitute of the +power of counting beyond 2. But such is the case; and in a few instances +languages have been found to be absolutely destitute of pure numeral words. +The Chiquitos of Bolivia had no real numerals whatever,[1] but expressed +their idea for "one" by the word _etama_, meaning alone. The Tacanas of the +same country have no numerals except those borrowed from Spanish, or from +Aymara or Peno, languages with which they have long been in contact.[2] A +few other South American languages are almost equally destitute of numeral +words. But even here, rudimentary as the number sense undoubtedly is, it is +not wholly lacking; and some indirect expression, or some form of +circumlocution, shows a conception of the difference between _one_ and +_two_, or at least, between _one_ and _many_. + +These facts must of necessity deter the mathematician from seeking to push +his investigation too far back toward the very origin of number. +Philosophers have endeavoured to establish certain propositions concerning +this subject, but, as might have been expected, have failed to reach any +common ground of agreement. Whewell has maintained that "such propositions +as that two and three make five are necessary truths, containing in them an +element of certainty beyond that which mere experience can give." Mill, on +the other hand, argues that any such statement merely expresses a truth +derived from early and constant experience; and in this view he is heartily +supported by Tylor.[3] But why this question should provoke controversy, it +is difficult for the mathematician to understand. Either view would seem to +be correct, according to the standpoint from which the question is +approached. We know of no language in which the suggestion of number does +not appear, and we must admit that the words which give expression to the +number sense would be among the early words to be formed in any language. +They express ideas which are, at first, wholly concrete, which are of the +greatest possible simplicity, and which seem in many ways to be clearly +understood, even by the higher orders of the brute creation. The origin of +number would in itself, then, appear to lie beyond the proper limits of +inquiry; and the primitive conception of number to be fundamental with +human thought. + +In connection with the assertion that the idea of number seems to be +understood by the higher orders of animals, the following brief quotation +from a paper by Sir John Lubbock may not be out of place: "Leroy ... +mentions a case in which a man was anxious to shoot a crow. 'To deceive +this suspicious bird, the plan was hit upon of sending two men to the watch +house, one of whom passed on, while the other remained; but the crow +counted and kept her distance. The next day three went, and again she +perceived that only two retired. In fine, it was found necessary to send +five or six men to the watch house to put her out in her calculation. The +crow, thinking that this number of men had passed by, lost no time in +returning.' From this he inferred that crows could count up to four. +Lichtenberg mentions a nightingale which was said to count up to three. +Every day he gave it three mealworms, one at a time. When it had finished +one it returned for another, but after the third it knew that the feast was +over.... There is an amusing and suggestive remark in Mr. Galton's +interesting _Narrative of an Explorer in Tropical South Africa_. After +describing the Demara's weakness in calculations, he says: 'Once while I +watched a Demara floundering hopelessly in a calculation on one side of me, +I observed, "Dinah," my spaniel, equally embarrassed on the other; she was +overlooking half a dozen of her new-born puppies, which had been removed +two or three times from her, and her anxiety was excessive, as she tried to +find out if they were all present, or if any were still missing. She kept +puzzling and running her eyes over them backwards and forwards, but could +not satisfy herself. She evidently had a vague notion of counting, but the +figure was too large for her brain. Taking the two as they stood, dog and +Demara, the comparison reflected no great honour on the man....' According +to my bird-nesting recollections, which I have refreshed by more recent +experience, if a nest contains four eggs, one may safely be taken; but if +two are removed, the bird generally deserts. Here, then, it would seem as +if we had some reason for supposing that there is sufficient intelligence +to distinguish three from four. An interesting consideration arises with +reference to the number of the victims allotted to each cell by the +solitary wasps. One species of Ammophila considers one large caterpillar of +_Noctua segetum_ enough; one species of Eumenes supplies its young with +five victims; another 10, 15, and even up to 24. The number appears to be +constant in each species. How does the insect know when her task is +fulfilled? Not by the cell being filled, for if some be removed, she does +not replace them. When she has brought her complement she considers her +task accomplished, whether the victims are still there or not. How, then, +does she know when she has made up the number 24? Perhaps it will be said +that each species feels some mysterious and innate tendency to provide a +certain number of victims. This would, under no circumstances, be any +explanation; but it is not in accordance with the facts. In the genus +Eumenes the males are much smaller than the females.... If the egg is male, +she supplies five; if female, 10 victims. Does she count? Certainly this +seems very like a commencement of arithmetic."[4] + +Many writers do not agree with the conclusions which Lubbock reaches; +maintaining that there is, in all such instances, a perception of greater +or less quantity rather than any idea of number. But a careful +consideration of the objections offered fails entirely to weaken the +argument. Example after example of a nature similar to those just quoted +might be given, indicating on the part of animals a perception of the +difference between 1 and 2, or between 2 and 3 and 4; and any reasoning +which tends to show that it is quantity rather than number which the animal +perceives, will apply with equal force to the Demara, the Chiquito, and the +Australian. Hence the actual origin of number may safely be excluded from +the limits of investigation, and, for the present, be left in the field of +pure speculation. + +A most inviting field for research is, however, furnished by the primitive +methods of counting and of giving visible expression to the idea of number. +Our starting-point must, of course, be the sign language, which always +precedes intelligible speech; and which is so convenient and so expressive +a method of communication that the human family, even in its most highly +developed branches, never wholly lays it aside. It may, indeed, be stated +as a universal law, that some practical method of numeration has, in the +childhood of every nation or tribe, preceded the formation of numeral +words. + +Practical methods of numeration are many in number and diverse in kind. But +the one primitive method of counting which seems to have been almost +universal throughout all time is the finger method. It is a matter of +common experience and observation that every child, when he begins to +count, turns instinctively to his fingers; and, with these convenient aids +as counters, tallies off the little number he has in mind. This method is +at once so natural and obvious that there can be no doubt that it has +always been employed by savage tribes, since the first appearance of the +human race in remote antiquity. All research among uncivilized peoples has +tended to confirm this view, were confirmation needed of anything so +patent. Occasionally some exception to this rule is found; or some +variation, such as is presented by the forest tribes of Brazil, who, +instead of counting on the fingers themselves, count on the joints of their +fingers.[5] As the entire number system of these tribes appears to be +limited to _three_, this variation is no cause for surprise. + +The variety in practical methods of numeration observed among savage races, +and among civilized peoples as well, is so great that any detailed account +of them would be almost impossible. In one region we find sticks or splints +used; in another, pebbles or shells; in another, simple scratches, or +notches cut in a stick, Robinson Crusoe fashion; in another, kernels or +little heaps of grain; in another, knots on a string; and so on, in +diversity of method almost endless. Such are the devices which have been, +and still are, to be found in the daily habit of great numbers of Indian, +negro, Mongolian, and Malay tribes; while, to pass at a single step to the +other extremity of intellectual development, the German student keeps his +beer score by chalk marks on the table or on the wall. But back of all +these devices, and forming a common origin to which all may be referred, is +the universal finger method; the method with which all begin, and which all +find too convenient ever to relinquish entirely, even though their +civilization be of the highest type. Any such mode of counting, whether +involving the use of the fingers or not, is to be regarded simply as an +extraneous aid in the expression or comprehension of an idea which the mind +cannot grasp, or cannot retain, without assistance. The German student +scores his reckoning with chalk marks because he might otherwise forget; +while the Andaman Islander counts on his fingers because he has no other +method of counting,--or, in other words, of grasping the idea of number. A +single illustration may be given which typifies all practical methods of +numeration. More than a century ago travellers in Madagascar observed a +curious but simple mode of ascertaining the number of soldiers in an +army.[6] Each soldier was made to go through a passage in the presence of +the principal chiefs; and as he went through, a pebble was dropped on the +ground. This continued until a heap of 10 was obtained, when one was set +aside and a new heap begun. Upon the completion of 10 heaps, a pebble was +set aside to indicate 100; and so on until the entire army had been +numbered. Another illustration, taken from the very antipodes of +Madagascar, recently found its way into print in an incidental manner,[7] +and is so good that it deserves a place beside de Flacourt's time-honoured +example. Mom Cely, a Southern negro of unknown age, finds herself in debt +to the storekeeper; and, unwilling to believe that the amount is as great +as he represents, she proceeds to investigate the matter in her own +peculiar way. She had "kept a tally of these purchases by means of a +string, in which she tied commemorative knots." When her creditor +"undertook to make the matter clear to Cely's comprehension, he had to +proceed upon a system of her own devising. A small notch was cut in a +smooth white stick for every dime she owed, and a large notch when the +dimes amounted to a dollar; for every five dollars a string was tied in the +fifth big notch, Cely keeping tally by the knots in her bit of twine; thus, +when two strings were tied about the stick, the ten dollars were seen to be +an indisputable fact." This interesting method of computing the amount of +her debt, whether an invention of her own or a survival of the African life +of her parents, served the old negro woman's purpose perfectly; and it +illustrates, as well as a score of examples could, the methods of +numeration to which the children of barbarism resort when any number is to +be expressed which exceeds the number of counters with which nature has +provided them. The fingers are, however, often employed in counting numbers +far above the first decade. After giving the Il-Oigob numerals up to 60, +Mueller adds:[8] "Above 60 all numbers, indicated by the proper figure +pantomime, are expressed by means of the word _ipi_." We know, moreover, +that many of the American Indian tribes count one ten after another on +their fingers; so that, whatever number they are endeavouring to indicate, +we need feel no surprise if the savage continues to use his fingers +throughout the entire extent of his counts. In rare instances we find +tribes which, like the Mairassis of the interior of New Guinea, appear to +use nothing but finger pantomime.[9] This tribe, though by no means +destitute of the number sense, is said to have no numerals whatever, but to +use the single word _awari_ with each show of fingers, no matter how few or +how many are displayed. + +In the methods of finger counting employed by savages a considerable degree +of uniformity has been observed. Not only does he use his fingers to assist +him in his tally, but he almost always begins with the little finger of his +left hand, thence proceeding towards the thumb, which is 5. From this point +onward the method varies. Sometimes the second 5 also is told off on the +left hand, the same order being observed as in the first 5; but oftener the +fingers of the right hand are used, with a reversal of the order previously +employed; _i.e._ the thumb denotes 6, the index finger 7, and so on to the +little finger, which completes the count to 10. + +At first thought there would seem to be no good reason for any marked +uniformity of method in finger counting. Observation among children fails +to detect any such thing; the child beginning, with almost entire +indifference, on the thumb or on the little finger of the left hand. My own +observation leads to the conclusion that very young children have a slight, +though not decided preference for beginning with the thumb. Experiments in +five different primary rooms in the public schools of Worcester, Mass., +showed that out of a total of 206 children, 57 began with the little finger +and 149 with the thumb. But the fact that nearly three-fourths of the +children began with the thumb, and but one-fourth with the little finger, +is really far less significant than would appear at first thought. Children +of this age, four to eight years, will count in either way, and sometimes +seem at a loss themselves to know where to begin. In one school room where +this experiment was tried the teacher incautiously asked one child to count +on his fingers, while all the other children in the room watched eagerly to +see what he would do. He began with the little finger--and so did every +child in the room after him. In another case the same error was made by the +teacher, and the child first asked began with the thumb. Every other child +in the room did the same, each following, consciously or unconsciously, the +example of the leader. The results from these two schools were of course +rejected from the totals which are given above; but they serve an excellent +purpose in showing how slight is the preference which very young children +have in this particular. So slight is it that no definite law can be +postulated of this age; but the tendency seems to be to hold the palm of +the hand downward, and then begin with the thumb. The writer once saw a boy +about seven years old trying to multiply 3 by 6; and his method of +procedure was as follows: holding his left hand with its palm down, he +touched with the forefinger of his right hand the thumb, forefinger, and +middle finger successively of his left hand. Then returning to his +starting-point, he told off a second three in the same manner. This process +he continued until he had obtained 6 threes, and then he announced his +result correctly. If he had been a few years older, he might not have +turned so readily to his thumb as a starting-point for any digital count. +The indifference manifested by very young children gradually disappears, +and at the age of twelve or thirteen the tendency is decidedly in the +direction of beginning with the little finger. Fully three-fourths of all +persons above that age will be found to count from the little finger toward +the thumb, thus reversing the proportion that was found to obtain in the +primary school rooms examined. + +With respect to finger counting among civilized peoples, we fail, then, to +find any universal law; the most that can be said is that more begin with +the little finger than with the thumb. But when we proceed to the study of +this slight but important particular among savages, we find them employing +a certain order of succession with such substantial uniformity that the +conclusion is inevitable that there must lie back of this some well-defined +reason, or perhaps instinct, which guides them in their choice. This +instinct is undoubtedly the outgrowth of the almost universal +right-handedness of the human race. In finger counting, whether among +children or adults, the beginning is made on the left hand, except in the +case of left-handed individuals; and even then the start is almost as +likely to be on the left hand as on the right. Savage tribes, as might be +expected, begin with the left hand. Not only is this custom almost +invariable, when tribes as a whole are considered, but the little finger is +nearly always called into requisition first. To account for this +uniformity, Lieutenant Gushing gives the following theory,[10] which is +well considered, and is based on the results of careful study and +observation among the Zuni Indians of the Southwest: "Primitive man when +abroad never lightly quit hold of his weapons. If he wanted to count, he +did as the Zuni afield does to-day; he tucked his instrument under his left +arm, thus constraining the latter, but leaving the right hand free, that he +might check off with it the fingers of the rigidly elevated left hand. From +the nature of this position, however, the palm of the left hand was +presented to the face of the counter, so that he had to begin his score on +the little finger of it, and continue his counting from the right leftward. +An inheritance of this may be detected to-day in the confirmed habit the +Zuni has of gesticulating from the right leftward, with the fingers of the +right hand over those of the left, whether he be counting and summing up, +or relating in any orderly manner." Here, then, is the reason for this +otherwise unaccountable phenomenon. If savage man is universally +right-handed, he will almost inevitably use the index finger of his right +hand to mark the fingers counted, and he will begin his count just where it +is most convenient. In his case it is with the little finger of the left +hand. In the case of the child trying to multiply 3 by 6, it was with the +thumb of the same hand. He had nothing to tuck under his arm; so, in +raising his left hand to a position where both eye and counting finger +could readily run over its fingers, he held the palm turned away from his +face. The same choice of starting-point then followed as with the +savage--the finger nearest his right hand; only in this case the finger was +a thumb. The deaf mute is sometimes taught in this manner, which is for him +an entirely natural manner. A left-handed child might be expected to count +in a left-to-right manner, beginning, probably, with the thumb of his right +hand. + +To the law just given, that savages begin to count on the little finger of +the left hand, there have been a few exceptions noted; and it has been +observed that the method of progression on the second hand is by no means +as invariable as on the first. The Otomacs[11] of South America began their +count with the thumb, and to express the number 3 would use the thumb, +forefinger, and middle finger. The Maipures,[12] oddly enough, seem to have +begun, in some cases at least, with the forefinger; for they are reported +as expressing 3 by means of the fore, middle, and ring fingers. The +Andamans[13] begin with the little finger of either hand, tapping the nose +with each finger in succession. If they have but one to express, they use +the forefinger of either hand, pronouncing at the same time the proper +word. The Bahnars,[14] one of the native tribes of the interior of Cochin +China, exhibit no particular order in the sequence of fingers used, though +they employ their digits freely to assist them in counting. Among certain +of the negro tribes of South Africa[15] the little finger of the right hand +is used for 1, and their count proceeds from right to left. With them, 6 is +the thumb of the left hand, 7 the forefinger, and so on. They hold the palm +downward instead of upward, and thus form a complete and striking exception +to the law which has been found to obtain with such substantial uniformity +in other parts of the uncivilized world. In Melanesia a few examples of +preference for beginning with the thumb may also be noticed. In the Banks +Islands the natives begin by turning down the thumb of the right hand, and +then the fingers in succession to the little finger, which is 5. This is +followed by the fingers of the left hand, both hands with closed fists +being held up to show the completed 10. In Lepers' Island, they begin with +the thumb, but, having reached 5 with the little finger, they do not pass +to the other hand, but throw up the fingers they have turned down, +beginning with the forefinger and keeping the thumb for 10.[16] In the use +of the single hand this people is quite peculiar. The second 5 is almost +invariably told off by savage tribes on the second hand, though in passing +from the one to the other primitive man does not follow any invariable law. +He marks 6 with either the thumb or the little finger. Probably the former +is the more common practice, but the statement cannot be made with any +degree of certainty. Among the Zulus the sequence is from thumb to thumb, +as is the case among the other South African tribes just mentioned; while +the Veis and numerous other African tribes pass from thumb to little +finger. The Eskimo, and nearly all the American Indian tribes, use the +correspondence between 6 and the thumb; but this habit is by no means +universal. Respecting progression from right to left or left to right on +the toes, there is no general law with which the author is familiar. Many +tribes never use the toes in counting, but signify the close of the first +10 by clapping the hands together, by a wave of the right hand, or by +designating some object; after which the fingers are again used as before. + +One other detail in finger counting is worthy of a moment's notice. It +seems to have been the opinion of earlier investigators that in his passage +from one finger to the next, the savage would invariably bend down, or +close, the last finger used; that is, that the count began with the fingers +open and outspread. This opinion is, however, erroneous. Several of the +Indian tribes of the West[17] begin with the hand clenched, and open the +fingers one by one as they proceed. This method is much less common than +the other, but that it exists is beyond question. + +In the Muralug Island, in the western part of Torres Strait, a somewhat +remarkable method of counting formerly existed, which grew out of, and is +to be regarded as an extension of, the digital method. Beginning with the +little finger of the left hand, the natives counted up to 5 in the usual +manner, and then, instead of passing to the other hand, or repeating the +count on the same fingers, they expressed the numbers from 6 to 10 by +touching and naming successively the left wrist, left elbow, left shoulder, +left breast, and sternum. Then the numbers from 11 to 19 were indicated by +the use, in inverse order, of the corresponding portions of the right side, +arm, and hand, the little finger of the right hand signifying 19. The words +used were in each case the actual names of the parts touched; the same +word, for example, standing for 6 and 14; but they were never used in the +numerical sense unless accompanied by the proper gesture, and bear no +resemblance to the common numerals, which are but few in number. This +method of counting is rapidly dying out among the natives of the island, +and is at the present time used only by old people.[18] Variations on this +most unusual custom have been found to exist in others of the neighbouring +islands, but none were exactly similar to it. One is also reminded by it of +a custom[19] which has for centuries prevailed among bargainers in the +East, of signifying numbers by touching the joints of each other's fingers +under a cloth. Every joint has a special signification; and the entire +system is undoubtedly a development from finger counting. The buyer or +seller will by this method express 6 or 60 by stretching out the thumb and +little finger and closing the rest of the fingers. The addition of the +fourth finger to the two thus used signifies 7 or 70; and so on. "It is +said that between two brokers settling a price by thus snipping with the +fingers, cleverness in bargaining, offering a little more, hesitating, +expressing an obstinate refusal to go further, etc., are as clearly +indicated as though the bargaining were being carried on in words. + +The place occupied, in the intellectual development of man, by finger +counting and by the many other artificial methods of reckoning,--pebbles, +shells, knots, the abacus, etc.,--seems to be this: The abstract processes +of addition, subtraction, multiplication, division, and even counting +itself, present to the mind a certain degree of difficulty. To assist in +overcoming that difficulty, these artificial aids are called in; and, among +savages of a low degree of development, like the Australians, they make +counting possible. A little higher in the intellectual scale, among the +American Indians, for example, they are employed merely as an artificial +aid to what could be done by mental effort alone. Finally, among +semi-civilized and civilized peoples, the same processes are retained, and +form a part of the daily life of almost every person who has to do with +counting, reckoning, or keeping tally in any manner whatever. They are no +longer necessary, but they are so convenient and so useful that +civilization can never dispense with them. The use of the abacus, in the +form of the ordinary numeral frame, has increased greatly within the past +few years; and the time may come when the abacus in its proper form will +again find in civilized countries a use as common as that of five centuries +ago. + +In the elaborate calculating machines of the present, such as are used by +life insurance actuaries and others having difficult computations to make, +we have the extreme of development in the direction of artificial aid to +reckoning. But instead of appearing merely as an extraneous aid to a +defective intelligence, it now presents itself as a machine so complex that +a high degree of intellectual power is required for the mere grasp of its +construction and method of working. + + + + + +CHAPTER II. + +NUMBER SYSTEM LIMITS. + + +With respect to the limits to which the number systems of the various +uncivilized races of the earth extend, recent anthropological research has +developed many interesting facts. In the case of the Chiquitos and a few +other native races of Bolivia we found no distinct number sense at all, as +far as could be judged from the absence, in their language, of numerals in +the proper sense of the word. How they indicated any number greater than +_one_ is a point still requiring investigation. In all other known +instances we find actual number systems, or what may for the sake of +uniformity be dignified by that name. In many cases, however, the numerals +existing are so few, and the ability to count is so limited, that the term +_number system_ is really an entire misnomer. + +Among the rudest tribes, those whose mode of living approaches most nearly +to utter savagery, we find a certain uniformity of method. The entire +number system may consist of but two words, _one_ and _many_; or of three +words, _one_, _two_, _many_. Or, the count may proceed to 3, 4, 5, 10, 20, +or 100; passing always, or almost always, from the distinct numeral limit +to the indefinite _many_ or several, which serves for the expression of any +number not readily grasped by the mind. As a matter of fact, most races +count as high as 10; but to this statement the exceptions are so numerous +that they deserve examination in some detail. In certain parts of the +world, notably among the native races of South America, Australia, and many +of the islands of Polynesia and Melanesia, a surprising paucity of numeral +words has been observed. The Encabellada of the Rio Napo have but two +distinct numerals; _tey_, 1, and _cayapa_, 2.[20] The Chaco languages[21] +of the Guaycuru stock are also notably poor in this respect. In the Mbocobi +dialect of this language the only native numerals are _yna tvak_, 1, and +_yfioaca_, 2. The Puris[22] count _omi_, 1, _curiri_, 2, _prica_, many; and +the Botocudos[23] _mokenam_, 1, _uruhu_, many. The Fuegans,[24] supposed to +have been able at one time to count to 10, have but three +numerals,--_kaoueli_, 1, _compaipi_, 2, _maten_, 3. The Campas of Peru[25] +possess only three separate words for the expression of number,--_patrio_, +1, _pitteni_, 2, _mahuani_, 3. Above 3 they proceed by combinations, as 1 +and 3 for 4, 1 and 1 and 3 for 5. Counting above 10 is, however, entirely +inconceivable to them, and any number beyond that limit they indicate by +_tohaine_, many. The Conibos,[26] of the same region, had, before their +contact with the Spanish, only _atchoupre_, 1, and _rrabui_, 2; though they +made some slight progress above 2 by means of reduplication. The Orejones, +one of the low, degraded tribes of the Upper Amazon,[27] have no names for +number except _nayhay_, 1, _nenacome_, 2, _feninichacome_, 3, +_ononoeomere_, 4. In the extensive vocabularies given by Von Martins,[28] +many similar examples are found. For the Bororos he gives only _couai_, 1, +_maeouai_, 2, _ouai_, 3. The last word, with the proper finger pantomime, +serves also for any higher number which falls within the grasp of their +comprehension. The Guachi manage to reach 5, but their numeration is of the +rudest kind, as the following scale shows: _tamak_, 1, _eu-echo,_ 2, +_eu-echo-kailau,_ 3, _eu-echo-way,_ 4, _localau_, 5. The Carajas counted by +a scale equally rude, and their conception of number seemed equally vague, +until contact with the neighbouring tribes furnished them with the means of +going beyond their original limit. Their scale shows clearly the uncertain, +feeble number sense which is so marked in the interior of South America. It +contains _wadewo_, 1, _wadebothoa_, 2, _wadeboaheodo_, 3, _wadebojeodo_, +4, _wadewajouclay_, 5, _wadewasori_, 6, or many. + +Turning to the languages of the extinct, or fast vanishing, tribes of +Australia, we find a still more noteworthy absence of numeral expressions. +In the Gudang dialect[29] but two numerals are found--_pirman_, 1, and +_ilabiu_, 2; in the Weedookarry, _ekkamurda_, 1, and _kootera_, 2; and in +the Queanbeyan, _midjemban_, 1, and _bollan_, 2. In a score or more of +instances the numerals stop at 3. The natives of Keppel Bay count _webben_, +1, _booli_, 2, _koorel_, 3; of the Boyne River, _karroon_, 1, _boodla_, 2, +_numma_, 3; of the Flinders River, _kooroin_, 1, _kurto_, 2, _kurto +kooroin_, 3; at the mouth of the Norman River, _lum_, 1, _buggar_, 2, +_orinch_, 3; the Eaw tribe, _koothea_, 1, _woother_, 2, _marronoo_, 3; the +Moree, _mal_, 1, _boolar_, 2, _kooliba_, 3; the Port Essington,[30] _erad_, +1, _nargarick_, 2, _nargarickelerad_, 3; the Darnly Islanders,[31] _netat_, +1, _naes_, 2, _naesa netat_, 3; and so on through a long list of tribes +whose numeral scales are equally scanty. A still larger number of tribes +show an ability to count one step further, to 4; but beyond this limit the +majority of Australian and Tasmanian tribes do not go. It seems most +remarkable that any human being should possess the ability to count to 4, +and not to 5. The number of fingers on one hand furnishes so obvious a +limit to any of these rudimentary systems, that positive evidence is needed +before one can accept the statement. A careful examination of the numerals +in upwards of a hundred Australian dialects leaves no doubt, however, that +such is the fact. The Australians in almost all cases count by pairs; and +so pronounced is this tendency that they pay but little attention to the +fingers. Some tribes do not appear ever to count beyond 2--a single pair. +Many more go one step further; but if they do, they are as likely as not to +designate their next numeral as two-one, or possibly, one-two. If this step +is taken, we may or may not find one more added to it, thus completing the +second pair. Still, the Australian's capacity for understanding anything +which pertains to number is so painfully limited that even here there is +sometimes an indefinite expression formed, as many, heap, or plenty, +instead of any distinct numeral; and it is probably true that no Australian +language contains a pure, simple numeral for 4. Curr, the best authority on +this subject, believes that, where a distinct word for 4 is given, +investigators have been deceived in every case.[32] If counting is carried +beyond 4, it is always by means of reduplication. A few tribes gave +expressions for 5, fewer still for 6, and a very small number appeared able +to reach 7. Possibly the ability to count extended still further; but if +so, it consisted undoubtedly in reckoning one pair after another, without +any consciousness whatever of the sum total save as a larger number. + +The numerals of a few additional tribes will show clearly that all distinct +perception of number is lost as soon as these races attempt to count above +3, or at most, 4. The Yuckaburra[33] natives can go no further than +_wigsin_, 1, _bullaroo_, 2, _goolbora_, 3. Above here all is referred to as +_moorgha_, many. The Marachowies[34] have but three distinct +numerals,--_cooma_, 1, _cootera_, 2, _murra_, 3. For 4 they say _minna_, +many. At Streaky Bay we find a similar list, with the same words, _kooma_ +and _kootera_, for 1 and 2, but entirely different terms, _karboo_ and +_yalkata_ for 3 and many. The same method obtains in the Minnal Yungar +tribe, where the only numerals are _kain_, 1, _kujal_, 2, _moa_, 3, and +_bulla_, plenty. In the Pinjarra dialect we find _doombart_, 1, _gugal_, 2, +_murdine_, 3, _boola_, plenty; and in the dialect described as belonging to +"Eyre's Sand Patch," three definite terms are given--_kean_, 1, _koojal_, +2, _yalgatta_, 3, while a fourth, _murna_, served to describe anything +greater. In all these examples the fourth numeral is indefinite; and the +same statement is true of many other Australian languages. But more +commonly still we find 4, and perhaps 3 also, expressed by reduplication. +In the Port Mackay dialect[35] the latter numeral is compound, the count +being _warpur_, 1, _boolera_, 2, _boolera warpur_, 3. For 4 the term is not +given. In the dialect which prevailed between the Albert and Tweed +rivers[36] the scale appears as _yaburu_, 1, _boolaroo_, 2, _boolaroo +yaburu_, 3, and _gurul_ for 4 or anything beyond. The Wiraduroi[37] have +_numbai_, 1, _bula_, 2, _bula numbai_, 3, _bungu_, 4, or many, and _bungu +galan_ or _bian galan_, 5, or very many. The Kamilaroi[38] scale is still +more irregular, compounding above 4 with little apparent method. The +numerals are _mal_, 1, _bular_, 2, _guliba_, 3, _bular bular_, 4, _bular +guliba_, 5, _guliba guliba_, 6. The last two numerals show that 5 is to +these natives simply 2-3, and 6 is 3-3. For additional examples of a +similar nature the extended list of Australian scales given in Chapter V. +may be consulted. + +Taken as a whole, the Australian and Tasmanian tribes seem to have been +distinctly inferior to those of South America in their ability to use and +to comprehend numerals. In all but two or three cases the Tasmanians[39] +were found to be unable to proceed beyond 2; and as the foregoing examples +have indicated, their Australian neighbours were but little better off. In +one or two instances we do find Australian numeral scales which reach 10, +and perhaps we may safely say 20. One of these is given in full in a +subsequent chapter, and its structure gives rise to the suspicion that it +was originally as limited as those of kindred tribes, and that it underwent +a considerable development after the natives had come in contact with the +Europeans. There is good reason to believe that no Australian in his wild +state could ever count intelligently to 7.[40] + +In certain portions of Asia, Africa, Melanesia, Polynesia, and North +America, are to be found races whose number systems are almost and +sometimes quite as limited as are those of the South. American and +Australian tribes already cited, but nowhere else do we find these so +abundant as in the two continents just mentioned, where example after +example might be cited of tribes whose ability to count is circumscribed +within the narrowest limits. The Veddas[41] of Ceylon have but two +numerals, _ekkame[=i]_, 1, _dekkamei_, 2. Beyond this they count +_otameekai, otameekai, otameekai_, etc.; _i.e._ "and one more, and one +more, and one more," and so on indefinitely. The Andamans,[42] inhabitants +of a group of islands in the Bay of Bengal, are equally limited in their +power of counting. They have _ubatulda_, 1, and _ikporda_, 2; but they can +go no further, except in a manner similar to that of the Veddas. Above two +they proceed wholly by means of the fingers, saying as they tap the nose +with each successive finger, _anka_, "and this." Only the more intelligent +of the Andamans can count at all, many of them seeming to be as nearly +destitute of the number sense as it is possible for a human being to be. +The Bushmen[43] of South Africa have but two numerals, the pronunciation of +which can hardly be indicated without other resources than those of the +English alphabet. Their word for 3 means, simply, many, as in the case of +some of the Australian tribes. The Watchandies[44] have but two simple +numerals, and their entire number system is _cooteon_, 1, _utaura_, 2, +_utarra cooteoo_, 3, _atarra utarra_, 4. Beyond this they can only say, +_booltha_, many, and _booltha bat_, very many. Although they have the +expressions here given for 3 and 4, they are reluctant to use them, and +only do so when absolutely required. The natives of Lower California[45] +cannot count above 5. A few of the more intelligent among them understand +the meaning of 2 fives, but this number seems entirely beyond the +comprehension of the ordinary native. The Comanches, curiously enough, are +so reluctant to employ their number words that they appear to prefer finger +pantomime instead, thus giving rise to the impression which at one time +became current, that they had no numerals at all for ordinary counting. + +Aside from the specific examples already given, a considerable number of +sweeping generalizations may be made, tending to show how rudimentary the +number sense may be in aboriginal life. Scores of the native dialects of +Australia and South America have been found containing number systems but +little more extensive than those alluded to above. The negro tribes of +Africa give the same testimony, as do many of the native races of Central +America, Mexico, and the Pacific coast of the United States and Canada, the +northern part of Siberia, Greenland, Labrador, and the arctic archipelago. +In speaking of the Eskimos of Point Barrow, Murdoch[46] says: "It was not +easy to obtain any accurate information about the numeral system of these +people, since in ordinary conversation they are not in the habit of +specifying any numbers above five." Counting is often carried higher than +this among certain of these northern tribes, but, save for occasional +examples, it is limited at best. Dr. Franz Boas, who has travelled +extensively among the Eskimos, and whose observations are always of the +most accurate nature, once told the author that he never met an Eskimo who +could count above 15. Their numerals actually do extend much higher; and a +stray numeral of Danish origin is now and then met with, showing that the +more intelligent among them are able to comprehend numbers of much greater +magnitude than this. But as Dr. Boas was engaged in active work among them +for three years, we may conclude that the Eskimo has an arithmetic but +little more extended than that which sufficed for the Australians and the +forest tribes of Brazil. Early Russian explorers among the northern tribes +of Siberia noticed the same difficulty in ordinary, every-day reckoning +among the natives. At first thought we might, then, state it as a general +law that those races which are lowest in the scale of civilization, have +the feeblest number sense also; or in other words, the least possible power +of grasping the abstract idea of number. + +But to this law there are many and important exceptions. The concurrent +testimony of explorers seems to be that savage races possess, in the great +majority of cases, the ability to count at least as high as 10. This limit +is often extended to 20, and not infrequently to 100. Again, we find 1000 +as the limit; or perhaps 10,000; and sometimes the savage carries his +number system on into the hundreds of thousands or millions. Indeed, the +high limit to which some savage races carry their numeration is far more +worthy of remark than the entire absence of the number sense exhibited by +others of apparently equal intelligence. If the life of any tribe is such +as to induce trade and barter with their neighbours, a considerable +quickness in reckoning will be developed among them. Otherwise this power +will remain dormant because there is but little in the ordinary life of +primitive man to call for its exercise. + +In giving 1, 2, 3, 5, 10, or any other small number as a system limit, it +must not be overlooked that this limit mentioned is in all cases the limit +of the spoken numerals at the savage's command. The actual ability to count +is almost always, and one is tempted to say always, somewhat greater than +their vocabularies would indicate. The Bushman has no number word that will +express for him anything higher than 2; but with the assistance of his +fingers he gropes his way on as far as 10. The Veddas, the Andamans, the +Guachi, the Botocudos, the Eskimos, and the thousand and one other tribes +which furnish such scanty numeral systems, almost all proceed with more or +less readiness as far as their fingers will carry them. As a matter of +fact, this limit is frequently extended to 20; the toes, the fingers of a +second man, or a recount of the savage's own fingers, serving as a tale for +the second 10. Allusion is again made to this in a later chapter, where the +subject of counting on the fingers and toes is examined more in detail. + +In saying that a savage can count to 10, to 20, or to 100, but little idea +is given of his real mental conception of any except the smallest numbers. +Want of familiarity with the use of numbers, and lack of convenient means +of comparison, must result in extreme indefiniteness of mental conception +and almost entire absence of exactness. The experience of Captain +Parry,[47] who found that the Eskimos made mistakes before they reached 7, +and of Humboldt,[48] who says that a Chayma might be made to say that his +age was either 18 or 60, has been duplicated by all investigators who have +had actual experience among savage races. Nor, on the other hand, is the +development of a numeral system an infallible index of mental power, or of +any real approach toward civilization. A continued use of the trading and +bargaining faculties must and does result in a familiarity with numbers +sufficient to enable savages to perform unexpected feats in reckoning. +Among some of the West African tribes this has actually been found to be +the case; and among the Yorubas of Abeokuta[49] the extraordinary saying, +"You may seem very clever, but you can't tell nine times nine," shows how +surprisingly this faculty has been developed, considering the general +condition of savagery in which the tribe lived. There can be no doubt that, +in general, the growth of the number sense keeps pace with the growth of +the intelligence in other respects. But when it is remembered that the +Tonga Islanders have numerals up to 100,000, and the Tembus, the Fingoes, +the Pondos, and a dozen other South African tribes go as high as 1,000,000; +and that Leigh Hunt never could learn the multiplication table, one must +confess that this law occasionally presents to our consideration remarkable +exceptions. + +While considering the extent of the savage's arithmetical knowledge, of his +ability to count and to grasp the meaning of number, it may not be amiss to +ask ourselves the question, what is the extent of the development of our +own number sense? To what limit can we absorb the idea of number, with a +complete appreciation of the idea of the number of units involved in any +written or spoken quantity? Our perfect system of numeration enables us to +express without difficulty any desired number, no matter how great or how +small it be. But how much of actually clear comprehension does the number +thus expressed convey to the mind? We say that one place is 100 miles from +another; that A paid B 1000 dollars for a certain piece of property; that a +given city contains 10,000 inhabitants; that 100,000 bushels of wheat were +shipped from Duluth or Odessa on such a day; that 1,000,000 feet of lumber +were destroyed by the fire of yesterday,--and as we pass from the smallest +to the largest of the numbers thus instanced, and from the largest on to +those still larger, we repeat the question just asked; and we repeat it +with a new sense of our own mental limitation. The number 100 +unquestionably stands for a distinct conception. Perhaps the same may be +said for 1000, though this could not be postulated with equal certainty. +But what of 10,000? If that number of persons were gathered together into a +single hall or amphitheatre, could an estimate be made by the average +onlooker which would approximate with any degree of accuracy the size of +the assembly? Or if an observer were stationed at a certain point, and +10,000 persons were to pass him in single file without his counting them as +they passed, what sort of an estimate would he make of their number? The +truth seems to be that our mental conception of number is much more limited +than is commonly thought, and that we unconsciously adopt some new unit as +a standard of comparison when we wish to render intelligible to our minds +any number of considerable magnitude. For example, we say that A has a +fortune of $1,000,000. The impression is at once conveyed of a considerable +degree of wealth, but it is rather from the fact that that fortune +represents an annual income of $40,000 than, from the actual magnitude of +the fortune itself. The number 1,000,000 is, in itself, so greatly in +excess of anything that enters into our daily experience that we have but a +vague conception of it, except as something very great. We are not, after +all, so very much better off than the child who, with his arms about his +mother's neck, informs her with perfect gravity and sincerity that he +"loves her a million bushels." His idea is merely of some very great +amount, and our own is often but little clearer when we use the expressions +which are so easily represented by a few digits. Among the uneducated +portions of civilized communities the limit of clear comprehension of +number is not only relatively, but absolutely, very low. Travellers in +Russia have informed the writer that the peasants of that country have no +distinct idea of a number consisting of but a few hundred even. There is no +reason to doubt this testimony. The entire life of a peasant might be +passed without his ever having occasion to use a number as great as 500, +and as a result he might have respecting that number an idea less distinct +than a trained mathematician would have of the distance from the earth to +the sun. De Quincey[50] incidentally mentions this characteristic in +narrating a conversation which occurred while he was at Carnarvon, a little +town in Wales. "It was on this occasion," he says, "that I learned how +vague are the ideas of number in unpractised minds. 'What number of people +do you think,' I said to an elderly person, 'will be assembled this day at +Carnarvon?' 'What number?' rejoined the person addressed; 'what number? +Well, really, now, I should reckon--perhaps a matter of four million.' Four +millions of _extra_ people in little Carnarvon, that could barely find +accommodation (I should calculate) for an extra four hundred!" So the +Eskimo and the South American Indian are, after all, not so very far behind +the "elderly person" of Carnarvon, in the distinct perception of a number +which familiarity renders to us absurdly small. + + + + + +CHAPTER III. + +THE ORIGIN OF NUMBER WORDS. + + +In the comparison of languages and the search for primitive root forms, no +class of expressions has been subjected to closer scrutiny than the little +cluster of words, found in each language, which constitutes a part of the +daily vocabulary of almost every human being--the words with which we begin +our counting. It is assumed, and with good reason, that these are among the +earlier words to appear in any language; and in the mutations of human +speech, they are found to suffer less than almost any other portion of a +language. Kinship between tongues remote from each other has in many +instances been detected by the similarity found to exist among the +every-day words of each; and among these words one may look with a good +degree of certainty for the 1, 2, 3, etc., of the number scale. So fruitful +has been this line of research, that the attempt has been made, even, to +establish a common origin for all the races of mankind by means of a +comparison of numeral words.[51] But in this instance, as in so many others +that will readily occur to the mind, the result has been that the theory +has finally taken possession of the author and reduced him to complete +subjugation, instead of remaining his servant and submitting to the +legitimate results of patient and careful investigation. Linguistic +research is so full of snares and pitfalls that the student must needs +employ the greatest degree of discrimination before asserting kinship of +race because of resemblances in vocabulary; or even relationship between +words in the same language because of some chance likeness of form that may +exist between them. Probably no one would argue that the English and the +Babusesse of Central Africa were of the same primitive stock simply because +in the language of the latter _five atano_ means 5, and _ten kumi_ means +10.[52] But, on the other hand, many will argue that, because the German +_zehn_ means 10, and _zehen_ means toes, the ancestors of the Germans +counted on their toes; and that with them, 10 was the complete count of the +toes. It may be so. We certainly have no evidence with which to disprove +this; but, before accepting it as a fact, or even as a reasonable +hypothesis, we may be pardoned for demanding some evidence aside from the +mere resemblance in the form of the words. If, in the study of numeral +words, form is to constitute our chief guide, we must expect now and then +to be confronted with facts which are not easily reconciled with any pet +theory. + +The scope of the present work will admit of no more than a hasty +examination of numeral forms, in which only actual and well ascertained +meanings will be considered. But here we are at the outset confronted with +a class of words whose original meanings appear to be entirely lost. They +are what may be termed the numerals proper--the native, uncompounded words +used to signify number. Such words are the one, two, three, etc., of +English; the eins, zwei, drei, etc., of German; words which must at some +time, in some prehistoric language, have had definite meanings entirely +apart from those which they now convey to our minds. In savage languages it +is sometimes possible to detect these meanings, and thus to obtain +possession of the clue that leads to the development, in the barbarian's +rude mind, of a count scale--a number system. But in languages like those +of modern Europe, the pedigree claimed by numerals is so long that, in the +successive changes through which they have passed, all trace of their +origin seems to have been lost. + +The actual number of such words is, however, surprisingly small in any +language. In English we count by simple words only to 10. From this point +onward all our numerals except "hundred" and "thousand" are compounds and +combinations of the names of smaller numbers. The words we employ to +designate the higher orders of units, as million, billion, trillion, etc., +are appropriated bodily from the Italian; and the native words _pair_, +_tale_, _brace_, _dozen_, _gross_, and _score_, can hardly be classed as +numerals in the strict sense of the word. German possesses exactly the same +number of native words in its numeral scale as English; and the same may be +said of the Teutonic languages generally, as well as of the Celtic, the +Latin, the Slavonic, and the Basque. This is, in fact, the universal method +observed in the formation of any numeral scale, though the actual number of +simple words may vary. The Chiquito language has but one numeral of any +kind whatever; English contains twelve simple terms; Sanskrit has +twenty-seven, while Japanese possesses twenty-four, and the Chinese a +number almost equally great. Very many languages, as might be expected, +contain special numeral expressions, such as the German _dutzend_ and the +French _dizaine_; but these, like the English _dozen_ and _score_, are not +to be regarded as numerals proper. + +The formation of numeral words shows at a glance the general method in +which any number scale has been built up. The primitive savage counts on +his fingers until he has reached the end of one, or more probably of both, +hands. Then, if he wishes to proceed farther, some mark is made, a pebble +is laid aside, a knot tied, or some similar device employed to signify that +all the counters at his disposal have been used. Then the count begins +anew, and to avoid multiplication of words, as well as to assist the +memory, the terms already used are again resorted to; and the name by which +the first halting-place was designated is repeated with each new numeral. +Hence the thirteen, fourteen, fifteen, etc., which are contractions of the +fuller expressions three-and-ten, four-and-ten, five-and-ten, etc. The +specific method of combination may not always be the same, as witness the +_eighteen_, or eight-ten, in English, and _dix-huit,_ or ten-eight, in +French; _forty-five_, or four-tens-five, in English, and _fuenf und +vierzig_, or five and four tens in German. But the general method is the +same the world over, presenting us with nothing but local variations, which +are, relatively speaking, entirely unimportant. With this fact in mind, we +can cease to wonder at the small number of simple numerals in any language. +It might, indeed, be queried, why do any languages, English and German, for +example, have unusual compounds for 11 and 12? It would seem as though the +regular method of compounding should begin with 10 and 1, instead of 10 and +3, in any language using a system with 10 as a base. An examination of +several hundred numeral scales shows that the Teutonic languages are +somewhat exceptional in this respect. The words _eleven_ and _twelve_ are +undoubtedly combinations, but not in the same direct sense as _thirteen_, +_twenty-five_, etc. The same may be said of the French _onze_, _douze_, +_treize_, _quatorze_, _quinze_, and _seize_, which are obvious compounds, +but not formed in the same manner as the numerals above that point. Almost +all civilized languages, however, except the Teutonic, and practically all +uncivilized languages, begin their direct numeral combinations as soon as +they have passed their number base, whatever that may be. To give an +illustration, selected quite at random from among the barbarous tribes of +Africa, the Ki-Swahili numeral scale runs as follows:[53] + + 1. moyyi, + 2. mbiri, + 3. tato, + 4. ena, + 5. tano, + 6. seta, + 7. saba, + 8. nani, + 9. kenda, + 10. kumi, + 11. kumi na moyyi, + 12. kumi na mbiri, + 13. kumi na tato, + etc. + +The words for 11, 12, and 13, are seen at a glance to signify ten-and-one, +ten-and-two, ten-and-three, and the count proceeds, as might be inferred, +in a similar manner as far as the number system extends. Our English +combinations are a little closer than these, and the combinations found in +certain other languages are, in turn, closer than those of the English; as +witness the _once_, 11, _doce_, 12, _trece_, 13, etc., of Spanish. But the +process is essentially the same, and the law may be accepted as practically +invariable, that all numerals greater than the base of a system are +expressed by compound words, except such as are necessary to establish some +new order of unit, as hundred or thousand. + +In the scale just given, it will be noticed that the larger number precedes +the smaller, giving 10 + 1, 10 + 2, etc., instead of 1 + 10, 2 + 10, etc. +This seems entirely natural, and hardly calls for any comment whatever. But +we have only to consider the formation of our English "teens" to see that +our own method is, at its inception, just the reverse of this. Thirteen, +14, and the remaining numerals up to 19 are formed by prefixing the smaller +number to the base; and it is only when we pass 20 that we return to the +more direct and obvious method of giving precedence to the larger. In +German and other Teutonic languages the inverse method is continued still +further. Here 25 is _fuenf und zwanzig_, 5 and 20; 92 is _zwei und neunzig_, +2 and 90, and so on to 99. Above 100 the order is made direct, as in +English. Of course, this mode of formation between 20 and 100 is +permissible in English, where "five and twenty" is just as correct a form +as twenty-five. But it is archaic, and would soon pass out of the language +altogether, were it not for the influence of some of the older writings +which have had a strong influence in preserving for us many of older and +more essentially Saxon forms of expression. + +Both the methods described above are found in all parts of the world, but +what I have called the direct is far more common than the other. In +general, where the smaller number precedes the larger it signifies +multiplication instead of addition. Thus, when we say "thirty," _i.e._ +three-ten, we mean 3 x 10; just as "three hundred" means 3 x 100. When the +larger precedes the smaller, we must usually understand addition. But to +both these rules there are very many exceptions. Among higher numbers the +inverse order is very rarely used; though even here an occasional exception +is found. The Taensa Indians, for example, place the smaller numbers before +the larger, no matter how far their scale may extend. To say 1881 they make +a complete inversion of our own order, beginning with 1 and ending with +1000. Their full numeral for this is _yeha av wabki mar-u-wab mar-u-haki_, +which means, literally, 1 + 80 + 100 x 8 + 100 x 10.[54] Such exceptions +are, however, quite rare. + +One other method of combination, that of subtraction, remains to be +considered. Every student of Latin will recall at once the _duodeviginti_, +2 from 20, and _undeviginti_, 1 from 20, which in that language are the +regular forms of expression for 18 and 19. At first they seem decidedly +odd; but familiarity soon accustoms one to them, and they cease entirely to +attract any special attention. This principle of subtraction, which, in the +formation of numeral words, is quite foreign to the genius of English, is +still of such common occurrence in other languages that the Latin examples +just given cease to be solitary instances. + +The origin of numerals of this class is to be found in the idea of +reference, not necessarily to the last, but to the nearest, halting-point +in the scale. Many tribes seem to regard 9 as "almost 10," and to give it a +name which conveys this thought. In the Mississaga, one of the numerous +Algonquin languages, we have, for example, the word _cangaswi_, "incomplete +10," for 9.[55] In the Kwakiutl of British Columbia, 8 as well as 9 is +formed in this way; these two numbers being _matlguanatl_, 10 - 2, and +_nanema_, 10 - 1, respectively.[56] In many of the languages of British +Columbia we find a similar formation for 8 and 9, or for 9 alone. The same +formation occurs in Malay, resulting in the numerals _delapan_, 10 - 2, and +_sambilan_ 10 - 1.[57] In Green Island, one of the New Ireland group, these +become simply _andra-lua_, "less 2," and _andra-si_, "less 1."[58] In the +Admiralty Islands this formation is carried back one step further, and not +only gives us _shua-luea_, "less 2," and _shu-ri_, "less 1," but also makes +7 appear as _sua-tolu_, "less 3."[59] Surprising as this numeral is, it is +more than matched by the Ainu scale, which carries subtraction back still +another step, and calls 6, 10 - 4. The four numerals from 6 to 9 in this +scale are respectively, _iwa_, 10 - 4, _arawa_, 10 - 3, _tupe-san_, 10 - 2, +and _sinepe-san_, 10 - 1.[60] Numerous examples of this kind of formation +will be found in later chapters of this work; but they will usually be +found to occur in one or both of the numerals, 8 and 9. Occasionally they +appear among the higher numbers; as in the Maya languages, where, for +example, 99 years is "one single year lacking from five score years,"[61] +and in the Arikara dialects, where 98 and 99 are "5 men minus" and "5 men 1 +not."[62] The Welsh, Danish, and other languages less easily accessible +than these to the general student, also furnish interesting examples of a +similar character. + +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 Zuni scale, given by Lieutenant Gushing[68] leaves nothing to be +desired: + + 1. toepinte = taken to start with. + 2. kwilli = put down together with. + 3. ha'[=i] = the equally dividing finger. + 4. awite = all the fingers all but done with. + 5. oepte = the notched off. + +This finishes the list of original simple numerals, the Zuni stopping, or +"notching off," when he finishes the fingers of one hand. Compounding now +begins. + + 6. topalik'ya = another brought to add to the done with. + 7. kwillilik'ya = two brought to and held up with the rest. + 8. hailik'ye = three brought to and held up with the rest. + 9. tenalik'ya = all but all are held up with the rest. + 10. aestem'thila = all the fingers. + 11. aestem'thla topayae'thl'tona = all the fingers and another over + above held. + +The process of formation indicated in 11 is used in the succeeding numerals +up to 19. + + 20. kwillik'yenaestem'thlan = two times all the fingers. + 100. aessiaestem'thlak'ya = the fingers all the fingers. + 1000. aessiaestem'thlanak'yenaestem'thla = the fingers all the fingers + times all the fingers. + +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 Zuni, 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 Zuni 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: + + 1. atauzik. + 2. madro. + 3. pinasun. + 4. sisaman. + 5. tudlemut. + 6. atautyimin akbinigin [tudlimu(t)] = 5 and 1 on the next. + 7. madronin akbinigin = twice on the next. + 8. pinasunin akbinigin = three times on the next. + 9. kodlinotaila = that which has not its 10. + 10. kodlin = the upper part--_i.e._ the fingers. + 14. akimiaxotaityuna = I have not 15. + 15. akimia. [This seems to be a real numeral word.] + 20. inyuina = a man come to an end. + 25. inyuina tudlimunin akbinidigin = a man come to an end and 5 on the + next. + 30. inyuina kodlinin akbinidigin = a man come to an end and 10 on the + next. + 35. inyuina akimiamin aipalin = a man come to an end accompanied by 1 + fifteen times. + 40. madro inyuina = 2 men come to an end. + +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] + + 1. atausek. + 2. mardluk. + 3. pingasut. + 4. sisamat. + 5. tatdlimat. + 6. arfinek-atausek = to the other hand 1. + 7. arfinek-mardluk = to the other hand 2. + 8. arfinek-pingasut = to the other hand 3. + 9. arfinek-sisamat = to the other hand 4. + 10. kulit. + +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: + + 1. abba. + 2. biama. + 3. kabbuhin. + 4. bibiti. + 5. abbatekkabe = 1 hand. + 6. abbatiman = 1 of the other. + 7. biamattiman = 2 of the other. + 8. kabbuhintiman = 3 of the other. + 9. bibitiman = 4 of the other. + 10. biamantekabbe = 2 hands. + 11. abba kutihibena = 1 from the feet. + 20. abba lukku = hands feet. + +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 Zuni scale, the words are digital from the outset. + + 1. inl'are = the end is bent. + 2. nak'e = another is bent. + 3. t'are = the middle is bent. + 4. dinri = there are no more except this. + 5. se-sunla-re = the row on the hand. + 6. elkke-t'are = 3 from each side. + 7.{ t'a-ye-oyertan = there are still 3 of them. + { inl'as dinri = on one side there are 4 of them. + 8. elkke-dinri = 4 on each side. + 9. inl'a-ye-oyert'an = there is still 1 more. + 10. onernan = finished on each side. + 11. onernan inl'are ttcharidhel = 1 complete and 1. + 12. onernan nak'e ttcharidhel = 1 complete and 2, etc. + +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 Zuni, 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: + + 5. amgnaitone = 1 hand complete. + 6. itacono amgna pona tevinitpe = 1 on the other hand. + 10. amgna aceponare = all of the 2 hands. + 11. puitta pona tevinitpe = 1 on the foot. + 16. itacono puitta pona tevinitpe = 1 on the other foot. + 20. tevin itoto = 1 man. + 21. itacono itoto jamgnar bona tevinitpe = 1 on the hands of another + man. + +In the Guarani[75] language of Paraguay the same method is found, with a +different form of expression for 20. Here the numerals in question are + + 5. asepopetei = one hand. + 10. asepomokoi = two hands. + 20. asepo asepi abe = hands and feet. + +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 + + 5. mi biche misa = one hand. + 10. mikriba misa sai = both hands. + 20. mikriba misa idecho ibi sai = both hands together with the feet. + +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 are[77] + + 1. tsomara. + 2. gar. + 3. gadiok. + 4. gahagani. + 5. tsuena yimana-ite = ended 1 hand. + 6. tsomara-hi = 1 on the other. + 7. gari-hi = 2 on the other. + 8. gadiog-ihi = 3 on the other. + 9. gahagani-hi = 4 on the other. + 10. tsuena yimana-die = ended both hands. + 11. tsomara yiri-tie = 1 on the foot. + 12. gar yiritie = 2 on the foot. + 20. tsuena yiri-die = ended both feet. + +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 +Maipures[78] of the Orinoco, who count + + 1. papita. + 2. avanume. + 3. apekiva. + 4. apekipaki. + 5. papitaerri capiti = 1 only hand. + 6. papita yana pauria capiti purena = 1 of the other hand we take. + 10. apanumerri capiti = 2 hands. + 11. papita yana kiti purena = 1 of the toes we take. + 20. papita camonee = 1 man. + 40. avanume camonee = 2 men. + 60. apekiva camonee = 3 men, etc. + +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 dialects[79] 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. + + 1. tey. + 2. cayapa. + 3. toazumba. + 4. cajezea = 2 with plural termination. + 5. teente = hand. + 6. teyentetey = hand + 1. + 7. teyente cayapa = hand + 2. + 8. teyente toazumba = hand + 3. + 9. teyente caesea = hand + 4. + 10. caya ente, or caya huena = 2 hands. + 11. caya ente-tey = 2 hands + 1. + 15. toazumba-ente = 3 hands. + 16. toazumba-ente-tey = 3 hands + 1. + 20. caesea ente = 4 hands. + +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: + + 1. mal. + 2. bular. + 3. guliba. + 4. bularbular = 2-2. + 5. mulanbu. + 6. malmulanbu mummi = 1 and 5 added on. + 7. bularmulanbu mummi = 2 and 5 added on. + 8. gulibamulanbu mummi = 3 and 5 added on. + 9. bularbularmulanbu mummi = 4 and 5 added on. + 10. bularin murra = belonging to the 2 hands. + 11. maldinna mummi = 1 of the toes added on (to the 10 fingers). + 12. bular dinna mummi = 2 of the toes added on. + 13. guliba dinna mummi = 3 of the toes added on. + 14. bular bular dinna mummi = 4 of the toes added on. + 15. mulanba dinna = 5 of the toes added on. + 16. mal dinna mulanbu = 1 and 5 toes. + 17. bular dinna mulanbu = 2 and 5 toes. + 18. guliba dinna mulanbu = 3 and 5 toes. + 19. bular bular dinna mulanbu = 4 and 5 toes. + 20. bularin dinna = belonging to the 2 feet. + +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] + + 1. nash, or nas. + 2. lap = hand. + 3. ndan. + 4. vunep = hand up. + 5. tunep = hand away. + 6. nadshkshapta = 1 I have bent over. + 7. lapkshapta = 2 I have bent over. + 8. ndankshapta = 3 I have bent over. + 9. nadshskeksh = 1 left over. + 10. taunep = hand hand? + +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 Zuni 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: + + 1. agit, or yaagit. + 2. uke. + 3. nipetuei. + 4. yepkatalet. + 5. isig-nisle-yaagit = hand fingers 1. + 6. isig-teet-yaagit = hand with 1. + 7. isig-teet-uke = hand with 2. + 8. isig-teet-nipetuei = hand with 3. + 9. isig-teet-yepkatalet = hand with 4. + 10. isig-uke-nisle = second hand fingers (lit. hand-two-fingers). + 11. isig-uke-nisle-teet-yaagit = second hand fingers with 1. + 20. isig-ape-nisle-lauel = hand foot fingers all. + +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: + + 8. kittukleemoot = middle finger. + 9. mikkeelukkamoot = fourth finger. + 10. eerkitkoka = little finger. + +Words of similar origin are found in the original Jiviro scale,[85] where +the native numerals are: + + 1. ala. + 2. catu. + 3. cala. + 4. encatu. + 5. alacoetegladu = 1 hand. + 6. intimutu = thumb (of second hand). + 7. tannituna = index finger. + 8. tannituna cabiasu = the finger next the index finger. + 9. bitin oetegla cabiasu = hand next to complete. + 10. catoegladu = 2 hands. + +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'alp[=e]eltk|gulbel | + | 3 |guant |guant |gutle |gulal |galtskan |galtskantk |guleont | + | 4 |tqalpq |tqalpq |tqalpq |tqalpqdal |tqaapskan |tqalpqsk |tqalpqalont | + | 5 |kct[=o]nc|kct[=o]nc|kct[=o]nc|kcenecal |k'etoentskan|kct[=o]onsk |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 |kp[=e]el |kpal |kp[=e]etskan|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, s[=a] = 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. + + + + + +CHAPTER VI. + +THE QUINARY SYSTEM. + + +The origin of the quinary mode of counting has been discussed with some +fulness in a preceding chapter, and upon that question but little more need +be said. It is the first of the natural systems. When the savage has +finished his count of the fingers of a single hand, he has reached this +natural number base. At this point he ceases to use simple numbers, and +begins the process of compounding. By some one of the numerous methods +illustrated in earlier chapters, he passes from 5 to 10, using here the +fingers of his second hand. He now has two fives; and, just as we say +"twenty," _i.e._ two tens, he says "two hands," "the second hand finished," +"all the fingers," "the fingers of both hands," "all the fingers come to an +end," or, much more rarely, "one man." That is, he is, in one of the many +ways at his command, saying "two fives." At 15 he has "three hands" or "one +foot"; and at 20 he pauses with "four hands," "hands and feet," "both +feet," "all the fingers of hands and feet," "hands and feet finished," or, +more probably, "one man." All these modes of expression are strictly +natural, and all have been found in the number scales which were, and in +many cases still are, in daily use among the uncivilized races of mankind. + +In its structure the quinary is the simplest, the most primitive, of the +natural systems. Its base is almost always expressed by a word meaning +"hand," or by some equivalent circumlocution, and its digital origin is +usually traced without difficulty. A consistent formation would require the +expression of 10 by some phrase meaning "two fives," 15 by "three fives," +etc. Such a scale is the one obtained from the Betoya language, already +mentioned in Chapter III., where the formation of the numerals is purely +quinary, as the following indicate:[227] + + 5. teente = 1 hand. + 10. cayaente, or caya huena = 2 hands. + 15. toazumba-ente = 3 hands. + 20. caesa-ente = 4 hands. + +The same formation appears, with greater or less distinctness, in many of +the quinary scales already quoted, and in many more of which mention might +be made. Collecting the significant numerals from a few such scales, and +tabulating them for the sake of convenience of comparison, we see this +point clearly illustrated by the following: + + + TAMANAC. + + 5. amnaitone = 1 hand. + 10. amna atse ponare = 2 hands. + + + ARAWAK, GUIANA. + + 5. abba tekkabe = 1 hand. + 10. biamantekkabe = 2 hands. + + + JIVIRO. + + 5. alacoetegladu = 1 hand. + 10. catoegladu = 2 hands. + + + NIAM NIAM + + 5. biswe + 10. bauwe = 2d 5. + + + NENGONES + + 5. se dono = the end (of the fingers of 1 hand). + 10. rewe tubenine = 2 series (of fingers). + + + SESAKE.[228] + + 5. lima = hand. + 10. dua lima = 2 hands. + + + AMBRYM.[229] + + 5. lim = hand. + 10. ra-lim = 2 hands. + + + PAMA.[229] + + 5. e-lime = hand. + 10. ha-lua-lim = the 2 hands. + + + DINKA.[230] + + 5. wdyets. + 10. wtyer, or wtyar = 5 x 2. + + + BARI + + 5. kanat + 10. puoek = 5 + 5? + + + KANURI + + 5. ugu. + 10. megu = 2 x 5. + + + RIO NORTE AND SAN ANTONIO.[231] + + 5. juyopamauj. + 10. juyopamauj ajte = 5 x 2. + + + API.[232] + + 5. lima. + 10. lua-lima = 2 x 5. + + + ERROMANGO + + 5. suku-rim. + 10. nduru-lim = 2 x 5. + + + TLINGIT, BRITISH COLUMBIA.[233] + + 5. kedjin (from djin = hand). + 10. djinkat = both hands? + +Thus far the quinary formation is simple and regular; and in view of the +evidence with which these and similar illustrations furnish us, it is most +surprising to find an eminent authority making the unequivocal statement +that the number 10 is nowhere expressed by 2 fives[234]--that all tribes +which begin their count on a quinary base express 10 by a simple word. It +is a fact, as will be fully illustrated in the following pages, that +quinary number systems, when extended, usually merge into either the +decimal or the vigesimal. The result is, of course, a compound of two, and +sometimes of three, systems in one scale. A pure quinary or vigesimal +number system is exceedingly rare; but quinary scales certainly do exist in +which, as far as we possess the numerals, no trace of any other influence +appears. It is also to be noticed that some tribes, like the Eskimos of +Point Barrow, though their systems may properly be classed as mixed +systems, exhibit a decided preference for 5 as a base, and in counting +objects, divided into groups of 5, obtaining the sum in this way.[235] + +But the savage, after counting up to 10, often finds himself unconsciously +impelled to depart from his strict reckoning by fives, and to assume a new +basis of reference. Take, for example, the Zuni system, in which the first +2 fives are: + + 5. oepte = the notched off. + 10. astem'thla = all the fingers. + +It will be noticed that the Zuni does not say "two hands," or "the fingers +of both hands," but simply "all the fingers." The 5 is no longer prominent, +but instead the mere notion of one entire count of the fingers has taken +its place. The division of the fingers into two sets of five each is still +in his mind, but it is no longer the leading idea. As the count proceeds +further, the quinary base may be retained, or it may be supplanted by a +decimal or a vigesimal base. How readily the one or the other may +predominate is seen by a glance at the following numerals: + + + GALIBI.[236] + + 5. atoneigne oietonai = 1 hand. + 10. oia batoue = the other hand. + 20. poupoupatoret oupoume = feet and hands. + 40. opoupoume = twice the feet and hands. + + + GUARANI.[237] + + 5. ace popetei = 1 hand. + 10. ace pomocoi = 2 hands. + 20. acepo acepiabe = hands and feet. + + + FATE.[238] + + 5. lima = hand. + 10. relima = 2 hands. + 20. relima rua = (2 x 5) x 2. + + + KIRIRI + + 5. mibika misa = 1 hand. + 10. mikriba misa sai = both hands. + 20. mikriba nusa ideko ibi sai = both hands together with the feet. + + + ZAMUCO + + 5. tsuena yimana-ite = ended 1 hand. + 10. tsuena yimana-die = ended both hands. + 20. tsuena yiri-die = ended both feet. + + + PIKUMBUL + + 5. mulanbu. + 10. bularin murra = belonging to the two hands. + 15. mulanba dinna = 5 toes added on (to the 10 fingers). + 20. bularin dinna = belonging to the 2 feet. + + + YARUROS.[239] + + 5. kani-iktsi-mo = 1 hand alone. + 10. yowa-iktsi-bo = all the hands. + 15. kani-tao-mo = 1 foot alone. + 20. kani-pume = 1 man. + +By the time 20 is reached the savage has probably allowed his conception of +any aggregate to be so far modified that this number does not present +itself to his mind as 4 fives. It may find expression in some phraseology +such as the Kiriris employ--"both hands together with the feet"--or in the +shorter "ended both feet" of the Zamucos, in which case we may presume that +he is conscious that his count has been completed by means of the four sets +of fives which are furnished by his hands and feet. But it is at least +equally probable that he instinctively divides his total into 2 tens, and +thus passes unconsciously from the quinary into the decimal scale. Again, +the summing up of the 10 fingers and 10 toes often results in the concept +of a single whole, a lump sum, so to speak, and the savage then says "one +man," or something that gives utterance to this thought of a new unit. This +leads the quinary into the vigesimal scale, and produces the combination so +often found in certain parts of the world. Thus the inevitable tendency of +any number system of quinary origin is toward the establishment of another +and larger base, and the formation of a number system in which both are +used. Wherever this is done, the greater of the two bases is always to be +regarded as the principal number base of the language, and the 5 as +entirely subordinate to it. It is hardly correct to say that, as a number +system is extended, the quinary element disappears and gives place to the +decimal or vigesimal, but rather that it becomes a factor of quite +secondary importance in the development of the scale. If, for example, 8 is +expressed by 5-3 in a quinary decimal system, 98 will be 9 x 10 + 5-3. The +quinary element does not disappear, but merely sinks into a relatively +unimportant position. + +One of the purest examples of quinary numeration is that furnished by the +Betoya scale, already given in full in Chapter III., and briefly mentioned +at the beginning of this chapter. In the simplicity and regularity of its +construction it is so noteworthy that it is worth repeating, as the first +of the long list of quinary systems given in the following pages. No +further comment is needed on it than that already made in connection with +its digital significance. As far as given by Dr. Brinton the scale is: + + 1. tey. + 2. cayapa. + 3. toazumba. + 4. cajezea = 2 with plural termination. + 5. teente = hand. + 6. teyente tey = hand 1. + 7. teyente cayapa = hand 2. + 8. teyente toazumba = hand 3. + 9. teyente caesea = hand 4. + 10. caya ente, or caya huena = 2 hands. + 11. caya ente-tey = 2 hands 1. + 15. toazumba-ente = 3 hands. + 16. toazumba-ente-tey = 3 hands 1. + 20. caesea ente = 4 hands. + +A far more common method of progression is furnished by languages which +interrupt the quinary formation at 10, and express that number by a single +word. Any scale in which this takes place can, from this point onward, be +quinary only in the subordinate sense to which allusion has just been made. +Examples of this are furnished in a more or less perfect manner by nearly +all so-called quinary-vigesimal and quinary-decimal scales. As fairly +representing this phase of number-system structure, I have selected the +first 20 numerals from the following languages: + + + WELSH.[240] + + 1. un. + 2. dau. + 3. tri. + 4. pedwar. + 5. pump. + 6. chwech. + 7. saith. + 8. wyth. + 9. naw. + 10. deg. + 11. un ar ddeg = 1 + 10. + 12. deuddeg = 2 + 10. + 13. tri ar ddeg = 3 + 10. + 14. pedwar ar ddeg = 4 + 10. + 15. pymtheg = 5 + 10. + 16. un ar bymtheg = 1 + 5 + 10. + 17. dau ar bymtheg = 2 + 5 + 10. + 18. tri ar bymtheg = 3 + 5 + 10. + 19. pedwar ar bymtheg = 4 + 5 + 10. + 20. ugain. + + + NAHUATL.[241] + + 1. ce. + 2. ome. + 3. yei. + 4. naui. + 5. macuilli. + 6. chiquacen = [5] + 1. + 7. chicome = [5] + 2. + 8. chicuey = [5] + 3. + 9. chiucnaui = [5] + 4. + 10. matlactli. + 11. matlactli oce = 10 + 1. + 12. matlactli omome = 10 + 2. + 13. matlactli omey = 10 + 3. + 14. matlactli onnaui = 10 + 4. + 15. caxtolli. + 16. caxtolli oce = 15 + 1. + 17. caxtolli omome = 15 + 2. + 18. caxtolli omey = 15 + 3. + 19. caxtolli onnaui = 15 + 4. + 20. cempualli = 1 account. + + + CANAQUE[242] NEW CALEDONIA. + + 1. chaguin. + 2. carou. + 3. careri. + 4. caboue + 5. cani. + 6. cani-mon-chaguin = 5 + 1. + 7. cani-mon-carou = 5 + 2. + 8. cani-mon-careri = 5 + 3. + 9. cani-mon-caboue = 5 + 4. + 10. panrere. + 11. panrere-mon-chaguin = 10 + 1. + 12. panrere-mon-carou = 10 + 2. + 13. panrere-mon-careri = 10 + 3. + 14. panrere-mon-caboue = 10 + 4. + 15. panrere-mon-cani = 10 + 5. + 16. panrere-mon-cani-mon-chaguin = 10 + 5 + 1. + 17. panrere-mon-cani-mon-carou = 10 + 5 + 2. + 18. panrere-mon-cani-mon-careri = 10 + 5 + 3. + 19. panrere-mon-cani-mon-caboue = 10 + 5 + 4. + 20. jaquemo = 1 person. + + + GUATO.[243] + + 1. cenai. + 2. dououni. + 3. coum. + 4. dekai. + 5. quinoui. + 6. cenai-caicaira = 1 on the other? + 7. dououni-caicaira = 2 on the other? + 8. coum-caicaira = 3 on the other? + 9. dekai-caicaira = 4 on the other? + 10. quinoi-da = 5 x 2. + 11. cenai-ai-caibo = 1 + (the) hands. + 12. dououni-ai-caibo = 2 + 10. + 13. coum-ai-caibo = 3 + 10. + 14. dekai-ai-caibo = 4 + 10. + 15. quin-oibo = 5 x 3. + 16. cenai-ai-quacoibo = 1 + 15. + 17. dououni-ai-quacoibo = 2 + 15. + 18. coum-ai-quacoibo = 3 + 15. + 19. dekai-ai-quacoibo = 4 + 15. + 20. quinoui-ai-quacoibo = 5 + 15. + +The meanings assigned to the numerals 6 to 9 are entirely conjectural. They +obviously mean 1, 2, 3, 4, taken a second time, and as the meanings I have +given are often found in primitive systems, they have, at a venture, been +given here. + + + LIFU, LOYALTY ISLANDS.[244] + + 1. ca. + 2. lue. + 3. koeni. + 4. eke. + 5. tji pi. + 6. ca ngemen = 1 above. + 7. lue ngemen = 2 above. + 8. koeni ngemen = 3 above. + 9. eke ngemen = 4 above. + 10. lue pi = 2 x 5. + 11. ca ko. + 12. lue ko. + 13. koeni ko. + 14. eke ko. + 15. koeni pi = 3 x 5. + 16. ca huai ano. + 17. lua huai ano. + 18. koeni huai ano. + 19. eke huai ano. + 20. ca atj = 1 man. + + + BONGO.[245] + + 1. kotu. + 2. ngorr. + 3. motta. + 4. neheo. + 5. mui. + 6. dokotu = [5] + 1. + 7. dongorr = [5] + 2. + 8. domotta = [5] + 3. + 9. doheo = [5] + 4. + 10. kih. + 11. ki dokpo kotu = 10 + 1. + 12. ki dokpo ngorr = 10 + 2. + 13. ki dokpo motta = 10 + 3. + 14. ki dokpo neheo = 10 + 4. + 15. ki dokpo mui = 10 + 5. + 16. ki dokpo mui do mui okpo kotu = 10 + 5 more, to 5, 1 more. + 17. ki dokpo mui do mui okpo ngorr = 10 + 5 more, to 5, 2 more. + 18. ki dokpo mui do mui okpo motta = 10 + 5 more, to 5, 3 more. + 19. ki dokpo mui do mui okpo nehea = 10 + 5 more, to 5, 4 more. + 20. mbaba kotu. + +Above 20, the Lufu and the Bongo systems are vigesimal, so that they are, +as a whole, mixed systems. + +The Welsh scale begins as though it were to present a pure decimal +structure, and no hint of the quinary element appears until it has passed +15. The Nahuatl, on the other hand, counts from 5 to 10 by the ordinary +quinary method, and then appears to pass into the decimal form. But when 16 +is reached, we find the quinary influence still persistent; and from this +point to 20, the numeral words in both scales are such as to show that the +notion of counting by fives is quite as prominent as the notion of +referring to 10 as a base. Above 20 the systems become vigesimal, with a +quinary or decimal structure appearing in all numerals except multiples of +20. Thus, in Welsh, 36 is _unarbymtheg ar ugain_, 1 + 5 + 10 + 20; and in +Nahuatl the same number is _cempualli caxtolli oce_, 20 + 15 + 1. Hence +these and similar number systems, though commonly alluded to as vigesimal, +are really mixed scales, with 20 as their primary base. The Canaque scale +differs from the Nahuatl only in forming a compound word for 15, instead of +introducing a new and simple term. + +In the examples which follow, it is not thought best to extend the lists of +numerals beyond 10, except in special instances where the illustration of +some particular point may demand it. The usual quinary scale will be found, +with a few exceptions like those just instanced, to have the following +structure or one similar to it in all essential details: 1, 2, 3, 4, 5, +5-1, 5-2, 5-3, 5-4, 10, 10-1, 10-2, 10-3, 10-4, 10-5, 10-5-1, 10-5-2, +10-5-3, 10-5-4, 20. From these forms the entire system can readily be +constructed as soon as it is known whether its principal base is to be 10 +or 20. + +Turning first to the native African languages, I have selected the +following quinary scales from the abundant material that has been collected +by the various explorers of the "Dark Continent." In some cases the +numerals of certain tribes, as given by one writer, are found to differ +widely from the same numerals as reported by another. No attempt has been +made at comparison of these varying forms of orthography, which are usually +to be ascribed to difference of nationality on the part of the collectors. + + + FELOOPS.[246] + + 1. enory. + 2. sickaba, or cookaba. + 3. sisajee. + 4. sibakeer. + 5. footuck. + 6. footuck-enory = 5-1. + 7. footuck-cookaba = 5-2. + 8. footuck-sisajee = 5-3. + 9. footuck-sibakeer = 5-4. + 10. sibankonyen. + + + KISSI.[247] + + 1. pili. + 2. miu. + 3. nga. + 4. iol. + 5. nguenu. + 6. ngom-pum = 5-1. + 7. ngom-miu = 5-2. + 8. ngommag = 5-3. + 9. nguenu-iol = 5-4. + 10. to. + + + ASHANTEE.[248] + + 1. tah. + 2. noo. + 3. sah. + 4. nah. + 5. taw. + 6. torata = 5 + 1. + 7. toorifeenoo = 5 + 2. + 8. toorifeessa = 5 + 3. + 9. toorifeena = 5 + 4. + 10. nopnoo. + + + BASA.[249] + + 1. do. + 2. so. + 3. ta. + 4. hinye. + 5. hum. + 6. hum-le-do = 5 + 1. + 7. hum-le-so = 5 + 2. + 8. hum-le-ta = 5 + 3. + 9. hum-le-hinyo = 5 + 4. + 10. bla-bue. + + + JALLONKAS.[250] + + 1. kidding. + 2. fidding. + 3. sarra. + 4. nani. + 5. soolo. + 6. seni. + 7. soolo ma fidding = 5 + 2. + 8. soolo ma sarra = 5 + 3. + 9. soolo ma nani = 5 + 4. + 10. nuff. + + + KRU. + + 1. da-do. + 2. de-son. + 3. de-tan. + 4. de-nie. + 5. de-mu. + 6. dme-du = 5-1. + 7. ne-son = [5] + 2. + 8. ne-tan = [5] + 3. + 9. sepadu = 10 - 1? + 10. pua. + + + JALOFFS.[251] + + 1. wean. + 2. yar. + 3. yat. + 4. yanet. + 5. judom. + 6. judom-wean = 5-1. + 7. judom-yar = 5-2. + 8. judom-yat = 5-3. + 9. judom yanet = 5-4. + 10. fook. + + + GOLO.[252] + + 1. mbali. + 2. bisi. + 3. bitta. + 4. banda. + 5. zonno. + 6. tsimmi tongbali = 5 + 1. + 7. tsimmi tobisi = 5 + 2. + 8. tsimmi tobitta = 5 + 3. + 9. tsimmi to banda = 5 + 4. + 10. nifo. + + + FOULAH.[253] + + 1. go. + 2. deeddee. + 3. tettee. + 4. nee. + 5. jouee. + 6. jego = 5-1. + 7. jedeeddee = 5-2. + 8. je-tettee = 5-3. + 9. je-nee = 5-4. + 10. sappo. + + + SOUSSOU.[254] + + 1. keren. + 2. firing. + 3. sarkan. + 4. nani. + 5. souli. + 6. seni. + 7. solo-fere = 5-2. + 8. solo-mazarkan = 5 + 3. + 9. solo-manani = 5 + 4. + 10. fu. + + + BULLOM.[255] + + 1. bul. + 2. tin. + 3. ra. + 4. hyul. + 5. men. + 6. men-bul = 5-1. + 7. men-tin = 5-2. + 8. men-ra = 5-3. + 9. men-hyul = 5-4. + 10. won. + + + VEI.[256] + + 1. dondo. + 2. fera. + 3. sagba. + 4. nani. + 5. soru. + 6. sun-dondo = 5-1. + 7. sum-fera = 5-2. + 8. sun-sagba = 5-3. + 9. sun-nani = 5-4. + 10. tan. + + + DINKA.[257] + + 1. tok. + 2. rou. + 3. dyak. + 4. nuan. + 5. wdyets. + 6. wdetem = 5-1. + 7. wderou = 5-2. + 8. bet, bed = 5-3. + 9. wdenuan = 5-4. + 10. wtyer = 5 x 2. + + + TEMNE. + + 1. in. + 2. ran. + 3. sas. + 4. anle. + 5. tr-amat. + 6. tr-amat rok-in = 5 + 1. + 7. tr-amat de ran = 5 + 2. + 8. tr-amat re sas = 5 + 3. + 9. tr-amat ro n-anle = 5 + 4. + 10. tr-ofatr. + + + ABAKER.[258] + + 1. kili. + 2. bore. + 3. dotla. + 4. ashe. + 5. ini. + 6. im kili = 5-1. + 7. im-bone = 5-2. + 8. ini-dotta = 5-3. + 9. tin ashe = 5-4. + 10. chica. + + + BAGRIMMA.[259] + + 1. kede. + 2. sab. + 3. muta. + 4. so. + 5. mi. + 6. mi-ga = 5 + 1. + 7. tsidi. + 8. marta = 5 + 2. + 9. do-so = [5] + 3 + 10. duk-keme. + + + PAPAA.[260] + + 1. depoo. + 2. auwi. + 3. ottong. + 4. enne. + 5. attong. + 6. attugo. + 7. atjuwe = [5] + 2. + 8. attiatong = [5] + 3. + 9. atjeenne = [5] + 4. + 10. awo. + + + EFIK.[261] + + 1. kiet. + 2. iba. + 3. ita. + 4. inan. + 5. itiun. + 6. itio-kiet = 5-1. + 7. itia-ba = 5-2. + 8. itia-eta = 5-3. + 9. osu-kiet = 10 - 1? + 10. duup. + + + NUPE.[262] + + 1. nini. + 2. gu-ba. + 3. gu-ta. + 4. gu-ni. + 5. gu-tsun. + 6. gu-sua-yin = 5 + 1. + 7. gu-tua-ba = 5 + 2. + 8. gu-tu-ta = 5 + 3. + 9. gu-tua-ni = 5 + 4. + 10. gu-wo. + + + MOKKO.[263] + + 1. kiae. + 2. iba. + 3. itta. + 4. inan. + 5. uettin. + 6. itjueekee = 5 + 1. + 7. ittiaba = 5 + 2. + 8. itteiata = 5 + 3. + 9. huschukiet. + 10. bueb. + + + KANURI.[264] + + 1. tilo. + 2. ndi. + 3. yasge. + 4. dege. + 5. ugu. + 6. arasge = 5 + 1. + 7. tulur. + 8. wusge = 5 + 3. + 9. legar. + 10. megu = 2 x 5. + + + BININ.[265] + + 1. bo. + 2. be. + 3. la. + 4. nin. + 5. tang. + 6. tahu = 5 + 1? + 7. tabi = 5 + 2. + 8. tara = 5 + 3. + 9. ianin (tanin?) = 5 + 4? + 10. te. + + + KREDY.[266] + + 1. baia. + 2. rommu. + 3. totto. + 4. sosso. + 5. saya. + 6. yembobaia = [5] + 1. + 7. yemborommu = [5] + 2. + 8. yembototto = [5] + 3. + 9. yembososso = [5] + 4. + 10. puh. + + + HERERO.[267] + + 1. mue. + 2. vari. + 3. tatu. + 4. ne. + 5. tano. + 6. hambou-mue = [5] + 1. + 7. hambou-vari = [5] + 2. + 8. hambou-tatu = [5] + 3. + 9. hambou-ne = [5] + 4. + 10. + + + KI-YAU.[268] + + 1. jumo. + 2. wawiri. + 3. watatu. + 4. mcheche. + 5. msano. + 6. musano na jumo = 5 + 1. + 7. musano na wiri = 5 + 2. + 8. musano na watatu = 5 + 3. + 9. musano na mcheche = 5 + 4. + 10. ikumi. + + + FERNANDO PO.[269] + + 1. muli. + 2. mempa. + 3. meta. + 4. miene. + 5. mimito. + 6. mimito na muli = 5 + 1. + 7. mimito na mempa = 5 + 2. + 8. mimito na meta = 5 + 3. + 9. mimito na miene = 5 + 4. + 10. miemieu = 5-5? + + + KI-NYASSA + + 1. kimodzi. + 2. vi-wiri. + 3. vi-tatu. + 4. vinye. + 5. visano. + 6. visano na kimodzi = 5 + 1. + 7. visano na vi-wiri = 5 + 2. + 8. visano na vitatu = 5 + 3. + 9. visano na vinye = 5 + 4. + 10. chikumi. + + + BALENGUE.[270] + + 1. guevoho. + 2. ibare. + 3. raro. + 4. inai. + 5. itano. + 6. itano na guevoho = 5 + 1. + 7. itano na ibare = 5 + 2. + 8. itano na raro = 5 + 3. + 9. itano na inai = 5 + 4. + 10. ndioum, or nai-hinai. + + + KUNAMA.[271] + + 1. ella. + 2. bare. + 3. sadde. + 4. salle. + 5. kussume. + 6. kon-t'-ella = hand 1. + 7. kon-te-bare = hand 2. + 8. kon-te-sadde = hand 3. + 9. kon-te-salle = hand 4. + 10. kol-lakada. + + + GOLA.[272] + + 1. ngoumou. + 2. ntie. + 3. ntai. + 4. tina. + 5. nonon. + 6. diegoum = [5] + 1. + 7. dientie = [5] + 2. + 8. dietai = [5] + 3. + 9. dectina = [5] + 4. + 10. esia. + + + BAREA.[273] + + 1. doko + 2. arega. + 3. sane. + 4. sone. + 5. oita. + 6. data. + 7. dz-ariga = 5 + 2. + 8. dis-sena = 5 + 3. + 9. lefete-mada = without 10. + 10. lefek. + + + MATIBANI.[274] + + 1. mosa. + 2. pili. + 3. taru. + 4. teje. + 5. taru. + 6. tana mosa = 5-1. + 7. tana pili = 5-2. + 8. tana taru = 5-3. + 9. loco. + 10. loco nakege. + + + BONZE.[275] + + 1. tan. + 2. vele. + 3. daba. + 4. nani. + 5. lolou. + 6. maida = [5] + 1. + 7. maifile = [5] + 2. + 8. maishaba = [5] + 3. + 9. mainan = [5] + 4. + 10. bou. + + + MPOVI + + 1. moueta. + 2. bevali. + 3. betata. + 4. benai. + 5. betani. + 6. betani moueta = 5-1. + 7. betani bevali = 5-2. + 8. betani betata = 5-3. + 9. betani benai = 5-4. + 10. nchinia. + + + TRITON'S BAY, NEW QUINEA.[276] + + 1. samosi. + 2. roueti. + 3. tourou. + 4. faat. + 5. rimi. + 6. rim-samosi = 5-1. + 7. rim-roueti = 5-2. + 8. rim-tourou = 5-3. + 9. rim-faat = 5-4. + 10. outsia. + + + ENDE, OR FLORES.[277] + + 1. sa. + 2. zua. + 3. telu. + 4. wutu. + 5. lima = hand. + 6. lima-sa = 5-1, or hand 1. + 7. lima-zua = 5-2. + 8. rua-butu = 2 x 4? + 9. trasa = [10] - 1? + 10. sabulu. + + + MALLICOLO.[278] + + 1. tseekaee. + 2. ery. + 3. erei. + 4. ebats. + 5. ereem. + 6. tsookaee = [5] + 1. + 7. gooy = [5] + 2. + 8. hoorey = [5] + 3. + 9. goodbats = [5] + 4. + 10. senearn. + + + EBON, MARSHALL ISLANDS.[279] + + 1. iuwun. + 2. drud. + 3. chilu. + 4. emer. + 5. lailem. + 6. chilchinu = 5 + 1. + 7. chilchime = 5 + 2. + 8. twalithuk = [10] - 2. + 9. twahmejuwou = [10] - 1. + 10. iungou. + + + UEA, LOYALTY ISLAND.[280] + + 1. tahi. + 2. lua. + 3. tolu. + 4. fa. + 5. lima. + 6. tahi. + 7. lua. + 8. tolu. + 9. fa. + 10. lima. + + + UEA.[280]--[another dialect.] + + 1. hacha. + 2. lo. + 3. kuun. + 4. thack. + 5. thabumb. + 6. lo-acha = 2d 1. + 7. lo-alo = 2d 2. + 8. lo-kuun = 2d 3. + 9. lo-thack = 2d 4. + 10. lebenetee. + + + ISLE OF PINES.[281] + + 1. ta. + 2. bo. + 3. beti. + 4. beu. + 5. ta-hue. + 6. no-ta = 2d 1. + 7. no-bo = 2d 2. + 8. no-beti = 2d 3. + 9. no-beu = 2d 4. + 10. de-kau. + + + UREPARAPARA, BANKS ISLANDS.[282] + + 1. vo towa. + 2. vo ro. + 3. vo tol. + 4. vo vet. + 5. teveliem = 1 hand. + 6. leve jea = other 1. + 7. leve ro = other 2. + 8. leve tol = other 3. + 9. leve vet = other 4. + 10. sanowul = 2 sets. + + + MOTA, BANKS ISLANDS.[282] + + 1. tuwale. + 2. nirua. + 3. nitol. + 4. nivat. + 5. tavelima = 1 hand. + 6. laveatea = other 1. + 7. lavearua = other 2. + 8. laveatol = other 3. + 9. laveavat = other 4. + 10. sanavul = 2 sets. + + + NEW CALEDONIA.[283] + + 1. parai. + 2. paroo. + 3. parghen. + 4. parbai. + 5. panim. + 6. panim-gha = 5-1. + 7. panim-roo = 5-2. + 8. panim-ghen = 5-3. + 9. panim-bai = 5-4. + 10. parooneek. + + + YENGEN, NEW CAL.[284] + + 1. hets. + 2. heluk. + 3. heyen. + 4. pobits. + 5. nim = hand. + 6. nim-wet = 5-1. + 7. nim-weluk = 5-2. + 8. nim-weyen = 5-3. + 9. nim-pobit = 5-4. + 10. pain-duk. + + + ANEITEUM.[285] + + 1. ethi. + 2. ero. + 3. eseik. + 4. manohwan. + 5. nikman. + 6. nikman cled et ethi = 5 + 1. + 7. nikman cled et oro = 5 + 2. + 8. nikman cled et eseik = 5 + 3. + 9. nikman cled et manohwan = 5 + 4. + 10. nikman lep ikman = 5 + 5. + + + TANNA + + 1. riti. + 2. karu. + 3. kahar. + 4. kefa. + 5. krirum. + 6. krirum riti = 5-1. + 7. krirum karu = 5-2. + 8. krirum kahar? = 5-3. + 9. krirum kefa? = 5-4. + 10. ---- + + + EROMANGA + + 1. sai. + 2. duru. + 3. disil. + 4. divat. + 5. siklim = 1 hand. + 6. misikai = other 1? + 7. siklim naru = 5-2. + 8. siklim disil = 5-3. + 9. siklim mindivat = 5 + 4. + 10. narolim = 2 hands. + + + FATE, NEW HEB.[286] + + 1. iskei. + 2. rua. + 3. tolu. + 4. bate. + 5. lima = hand. + 6. la tesa = other 1. + 7. la rua = other 2. + 8. la tolu = other 3. + 9. la fiti = other 4. + 10. relima = 2 hands. + + + API, NEW HEB. + + 1. tai. + 2. lua. + 3. tolu. + 4. vari. + 5. lima = hand. + 6. o rai = other 1. + 7. o lua = other 2. + 8. o tolo = other 3. + 9. o vari = other 4. + 10. lua lima = 2 hands. + + + SESAKE, NEW HEB. + + 1. sikai. + 2. dua. + 3. dolu. + 4. pati. + 5. lima = hand. + 6. la tesa = other 1. + 7. la dua = other 2. + 8. la dolu = other 3. + 9. lo veti = other 4. + 10. dua lima = 2 hands. + + + PAMA, NEW HEB. + + 1. tai. + 2. e lua. + 3. e tolu. + 4. e hati. + 5. e lime = hand. + 6. a hitai = other 1. + 7. o lu = other 2. + 8. o tolu = other 3. + 9. o hati = other 4. + 10. ha lua lim = 2 hands + + + AURORA, NEW HEB. + + 1. tewa. + 2. i rua. + 3. i tol. + 4. i vat. + 5. tavalima = 1 hand. + 6. lava tea = other 1. + 7. lava rua = other 2. + 8. lava tol = other 3. + 9. la vat = other 4. + 10. sanwulu = two sets. + + + TOBI.[287] + + 1. yat. + 2. glu. + 3. ya. + 4. uan. + 5. yanim = 1 hand. + 6. yawor = other 1. + 7. yavic = other 2. + 8. yawa = other 3. + 9. yatu = other 4. + 10. yasec. + + + PALM ISLAND.[288] + + 1. yonkol. + 2. yakka. + 3. tetjora. + 4. tarko. + 5. yonkol mala = 1 hand. + + + JAJOWERONG, VICTORIA.[288] + + 1. kiarp. + 2. bulaits. + 3. bulaits kiarp = 2-1. + 4. bulaits bulaits = 2-2. + 5. kiarp munnar = 1 hand. + 6. bulaits bulaits bulaits = 2-2-2. + 10. bulaits munnar = 2 hands. + +The last two scales deserve special notice. They are Australian scales, and +the former is strongly binary, as are so many others of that continent. But +both show an incipient quinary tendency in their names for 5 and 10. + + + CAMBODIA.[289] + + 1. muy. + 2. pir. + 3. bey. + 4. buon. + 5. pram. + 6. pram muy = 5-1. + 7. pram pil = 5-2. + 8. pram bey = 5-3. + 9. pram buon = 5-4. + 10. dap. + + + TSCHUKSCHI.[290] + + 1. inen. + 2. nirach. + 3. n'roch. + 4. n'rach. + 5. miligen = hand. + 6. inen miligen = 1-5. + 7. nirach miligen = 2-5. + 8. anwrotkin. + 9. chona tsinki. + 10. migitken = both hands. + + + KOTTISCH[291] + + 1. hutsa. + 2. ina. + 3. tona. + 4. sega. + 5. chega. + 6. chelutsa = 5 + 1. + 7. chelina = 5 + 2. + 8. chaltona = 5 + 3. + 9. tsumnaga = 10 - 1. + 10. haga. + + + ESKIMO OF N.-W. ALASKA.[292] + + 1. a towshek. + 2. hipah, or malho. + 3. pingishute. + 4. sesaimat. + 5. talema. + 6. okvinile, or ahchegaret = another 1? + 7. talema-malronik = 5-two of them. + 8. pingishu-okvingile = 2d 3? + 9. kolingotalia = 10 - 1? + 10. koleet. + + + KAMTSCHATKA, SOUTH.[293] + + 1. dischak. + 2. kascha. + 3. tschook. + 4. tschaaka. + 5. kumnaka. + 6. ky'lkoka. + 7. itatyk = 2 + 5. + 8. tschookotuk = 3 + 5. + 9. tschuaktuk = 4 + 5. + 10. kumechtuk = 5 + 5. + + + ALEUTS[294] + + 1. ataqan. + 2. aljak. + 3. qankun. + 4. sitsin. + 5. tsan = my hand. + 6. atun = 1 + 5. + 7. ulun = 2 + 5. + 8. qamtsin = 3 + 5. + 9. sitsin = 4 + 5. + 10. hatsiq. + + + TCHIGLIT, MACKENZIE R.[295] + + 1. ataotcirkr. + 2. aypak, or malloerok. + 3. illaak, or pinatcut. + 4. tcitamat. + 5. tallemat. + 6. arveneloerit. + 7. arveneloerit-aypak = 5 + 2. + 8. arveneloerit-illaak = 5 + 3. + 9. arveneloerit-tcitamat = 5 + 4. + 10. krolit. + + + SAHAPTIN (NEZ PERCES).[296] + + 1. naks. + 2. lapit. + 3. mitat. + 4. pi-lapt = 2 x 2. + 5. pachat. + 6. oi-laks = [5] + 1. + 7. oi-napt = [5] + 2. + 8. oi-matat = [5] + 3. + 9. koits. + 10. putimpt. + + + GREENLAND.[297] + + 1. atauseq. + 2. machdluq. + 3. pinasut. + 4. sisamat + 5. tadlimat. + 6. achfineq-atauseq = other hand 1. + 7. achfineq-machdluq = other hand 2. + 8. achfineq-pinasut = other hand 3. + 9. achfineq-sisamat = other hand 4. + 10. qulit. + 11. achqaneq-atauseq = first foot 1. + 12. achqaneq-machdluq = first foot 2. + 13. achqaneq-pinasut = first foot 3. + 14. achqaneq-sisamat = first foot 4. + 15. achfechsaneq? + 16. achfechsaneq-atauseq = other foot 1. + 17. achfechsaneq-machdlup = other foot 2. + 18. achfechsaneq-pinasut = other foot 3. + 19. achfechsaneq-sisamat = other foot 4. + 20. inuk navdlucho = a man ended. + +Up to this point the Greenlander's scale is almost purely quinary. Like +those of which mention was made at the beginning of this chapter, it +persists in progressing by fives until it reaches 20, when it announces a +new base, which shows that the system will from now on be vigesimal. This +scale is one of the most interesting of which we have any record, and will +be noticed again in the next chapter. In many respects it is like the scale +of the Point Barrow Eskimo, which was given early in Chapter III. The +Eskimo languages are characteristically quinary-vigesimal in their number +systems, but few of them present such perfect examples of that method of +counting as do the two just mentioned. + + + CHIPPEWAY.[298] + + 1. bejig. + 2. nij. + 3. nisswi. + 4. niwin. + 5. nanun. + 6. ningotwasswi = 1 again? + 7. nijwasswi = 2 again? + 8. nishwasswi = 3 again? + 9. jangasswi = 4 again? + 10. midasswi = 5 again. + + + MASSACHUSETTS.[299] + + 1. nequt. + 2. neese. + 3. nish. + 4. yaw. + 5. napanna = on one side, _i.e._ 1 hand. + 6. nequttatash = 1 added. + 7. nesausuk = 2 again? + 8. shawosuk = 3 again? + 9. pashoogun = it comes near, _i.e._ to 10. + 10. puik. + + + OJIBWA OF CHEGOIMEGON.[300] + + 1. bashik. + 2. neensh. + 3. niswe. + 4. newin. + 5. nanun. + 6. ningodwaswe = 1 again? + 7. nishwaswe = 2 again? + 8. shouswe = 3 again? + 9. shangaswe = 4 again? + 10. medaswe = 5 again? + + + OTTAWA. + + 1. ningotchau. + 2. ninjwa. + 3. niswa. + 4. niwin. + 5. nanau. + 6. ningotwaswi = 1 again? + 7. ninjwaswi = 2 again? + 8. nichwaswi = 3 again? + 9. shang. + 10. kwetch. + + + DELAWARE. + + 1. n'gutti. + 2. niskha. + 3. nakha. + 4. newa. + 5. nalan [akin to palenach, hand]. + 6. guttash = 1 on the other side. + 7. nishash = 2 on the other side. + 8. khaash = 3 on the other side. + 9. peshgonk = coming near. + 10. tellen = no more. + + + SHAWNOE. + + 1. negote. + 2. neshwa. + 3. nithuie. + 4. newe. + 5. nialinwe = gone. + 6. negotewathwe = 1 further. + 7. neshwathwe = 2 further. + 8. sashekswa = 3 further? + 9. chakatswe [akin to chagisse, "used up"]. + 10. metathwe = no further. + + + MICMAC.[301] + + 1. naiookt. + 2. tahboo. + 3. seest. + 4. naioo. + 5. nahn. + 6. usoo-cum. + 7. eloo-igunuk. + 8. oo-gumoolchin. + 9. pescoonaduk. + 10. mtlin. + +One peculiarity of the Micmac numerals is most noteworthy. The numerals are +real verbs, instead of adjectives, or, as is sometimes the case, nouns. +They are conjugated through all the variations of mood, tense, person, and +number. The forms given above are not those that would be used in counting, +but are for specific use, being varied according to the thought it was +intended to express. For example, _naiooktaich_ = there is 1, is present +tense; _naiooktaichcus_, there was 1, is imperfect; and _encoodaichdedou_, +there will be 1, is future. The variation in person is shown by the +following inflection: + + + PRESENT TENSE. + + 1st pers. tahboosee-ek = there are 2 of us. + 2d pers. tahboosee-yok = there are 2 of you. + 3d pers. tahboo-sijik = there are 2 of them. + + + IMPERFECT TENSE. + + 1st pers. tahboosee-egup = there were 2 of us. + 2d pers. tahboosee-yogup = there were 2 of you. + 3d pers. tahboosee-sibunik = there were 2 of them. + + + FUTURE TENSE. + + 3d pers. tahboosee-dak = there will be 2 of them, etc. + +The negative form is also comprehended in the list of possible variations. +Thus, _tahboo-seekw_, there are not 2 of them; _mah tahboo-seekw_, there +will not be 2 of them; and so on, through all the changes which the +conjugation of the verb permits. + + + OLD ALGONQUIN. + + 1. peygik. + 2. ninsh. + 3. nisswey. + 4. neyoo. + 5. nahran = gone. + 6. ningootwassoo = 1 on the other side. + 7. ninshwassoo = 2 on the other side. + 8. nisswasso = 3 on the other side. + 9. shangassoo [akin to chagisse, "used up"]. + 10. mitassoo = no further. + + + OMAHA. + + 1. meeachchee. + 2. nomba. + 3. rabeenee. + 4. tooba. + 5. satta = hand, _i.e._ all the fingers turned down. + 6. shappai = 1 more. + 7. painumba = fingers 2. + 8. pairabeenee = fingers 3. + 9. shonka = only 1 finger (remains). + 10. kraibaira = unbent.[302] + + + CHOCTAW. + + 1. achofee. + 2. tuklo. + 3. tuchina. + 4. ushta. + 5. tahlape = the first hand ends. + 6. hanali. + 7. untuklo = again 2. + 8. untuchina = again 3. + 9. chokali = soon the end; _i.e._ next the last. + 10. pokoli. + + + CADDOE. + + 1. kouanigh. + 2. behit. + 3. daho. + 4. hehweh. + 5. dihsehkon. + 6. dunkeh. + 7. bisekah = 5 + 2. + 8. dousehka = 5 + 3. + 9. hehwehsehka = 4 + hand. + 10. behnehaugh. + + + CHIPPEWAY. + + 1. payshik. + 2. neesh. + 3. neeswoy. + 4. neon. + 5. naman = gone. + 6. nequtwosswoy = 1 on the other side. + 7. neeshswosswoy = 2 on the other side. + 8. swoswoy = 3 on the other side? + 9. shangosswoy [akin to chagissi, "used up"]. + 10. metosswoy = no further. + + + ADAIZE. + + 1. nancas. + 2. nass. + 3. colle. + 4. tacache. + 5. seppacan. + 6. pacanancus = 5 + 1. + 7. pacaness = 5 + 2. + 8. pacalcon = 5 + 3. + 9. sickinish = hands minus? + 10. neusne. + + + PAWNEE. + + 1. askoo. + 2. peetkoo. + 3. touweet. + 4. shkeetiksh. + 5. sheeooksh = hands half. + 6. sheekshabish = 5 + 1. + 7. peetkoosheeshabish = 2 + 5. + 8. touweetshabish = 3 + 5. + 9. looksheereewa = 10 - 1. + 10. looksheeree = 2d 5? + + + MINSI. + + 1. gutti. + 2. niskha. + 3. nakba. + 4. newa. + 5. nulan = gone? + 6. guttash = 1 added. + 7. nishoash = 2 added. + 8. khaash = 3 added. + 9. noweli. + 10. wimbat. + + + KONLISCHEN. + + 1. tlek. + 2. tech. + 3. nezk. + 4. taakun. + 5. kejetschin. + 6. klet uschu = 5 + 1. + 7. tachate uschu = 5 + 2. + 8. nesket uschu = 5 + 3. + 9. kuschok = 10 - 1? + 10. tschinkat. + + + TLINGIT.[303] + + 1. tlek. + 2. deq. + 3. natsk. + 4. dak'on = 2d 2. + 5. kedjin = hand. + 6. tle durcu = other 1. + 7. daqa durcu = other 2. + 8. natska durcu = other 3. + 9. gocuk. + 10. djinkat = both hands. + + + RAPID, OR FALL, INDIANS. + + 1. karci. + 2. neece. + 3. narce. + 4. nean. + 5. yautune. + 6. neteartuce = 1 over? + 7. nesartuce = 2 over? + 8. narswartuce = 3 over? + 9. anharbetwartuce = 4 over? + 10. mettartuce = no further? + + + HEILTSUK.[304] + + 1. men. + 2. matl. + 3. yutq. + 4. mu. + 5. sky'a. + 6. katla. + 7. matlaaus = other 2? + 8. yutquaus = other 3? + 9. mamene = 10 - 1. + 10. aiky'as. + + + NOOTKA.[305] + + 1. nup. + 2. atla. + 3. katstsa. + 4. mo. + 5. sutca. + 6. nopo = other 1? + 7. atlpo = other 2? + 8. atlakutl = 10 - 2. + 9. ts'owakutl = 10 - 1. + 10. haiu. + + + TSIMSHIAN.[306] + + 1. gyak. + 2. tepqat. + 3. guant. + 4. tqalpq. + 5. kctonc (from _anon_, hand). + 6. kalt = 2d 1. + 7. t'epqalt = 2d 2. + 8. guandalt = 2d 3? + 9. kctemac. + 10. gy'ap. + + + BILQULA.[306] + + 1. (s)maotl. + 2. tlnos. + 3. asmost. + 4. mos. + 5. tsech. + 6. tqotl = 2d 1? + 7. nustlnos = 2d 2? + 8. k'etlnos = 2 x 4. + 9. k'esman. + 10. tskchlakcht. + + + MOLELE.[307] + + 1. mangu. + 2. lapku. + 3. mutka. + 4. pipa. + 5. pika. + 6. napitka = 1 + 5. + 7. lapitka = 2 + 5. + 8. mutpitka = 3 + 5. + 9. laginstshiatkus. + 10. nawitspu. + + + WAIILATPU.[308] + + 1. na. + 2. leplin. + 3. matnin. + 4. piping. + 5. tawit. + 6. noina = [5] + 1. + 7. noilip = [5] + 2. + 8. noimat = [5] + 3. + 9. tanauiaishimshim. + 10. ningitelp. + + + LUTUAMI.[307] + + 1. natshik. + 2. lapit. + 3. ntani. + 4. wonip. + 5. tonapni. + 6. nakskishuptane = 1 + 5. + 7. tapkishuptane = 2 + 5. + 8. ndanekishuptane = 3 + 5. + 9. natskaiakish = 10 - 1. + 10. taunip. + + + SASTE (SHASTA).[309] + + 1. tshiamu. + 2. hoka. + 3. hatski. + 4. irahaia. + 5. etsha. + 6. tahaia. + 7. hokaikinis = 2 + 5. + 8. hatsikikiri = 3 + 5. + 9. kirihariki-ikiriu. + 10. etsehewi. + + + CAHUILLO.[310] + + 1. supli. + 2. mewi. + 3. mepai. + 4. mewittsu. + 5. nomekadnun. + 6. kadnun-supli = 5-1. + 7. kan-munwi = 5-2. + 8. kan-munpa = 5-3. + 9. kan-munwitsu = 5-4. + 10. nomatsumi. + + + TIMUKUA.[311] + + 1. yaha. + 2. yutsa. + 3. hapu. + 4. tseketa. + 5. marua. + 6. mareka = 5 + 1 + 7. pikitsa = 5 + 2 + 8. pikinahu = 5 + 3 + 9. peke-tsaketa = 5 + 4 + 10. tuma. + + + OTOMI[312] + + 1. nara. + 2. yocho. + 3. chiu. + 4. gocho. + 5. kuto. + 6. rato = 1 + 5. + 7. yoto = 2 + 5. + 8. chiato = 3 + 5. + 9. guto = 4 + 5. + 10. reta. + + + TARASCO.[313] + + 1. ma. + 2. dziman. + 3. tanimo. + 4. tamu. + 5. yumu. + 6. kuimu. + 7. yun-dziman = [5] + 2. + 8. yun-tanimo = [5] + 3. + 9. yun-tamu = [5] + 4. + 10. temben. + + + MATLALTZINCAN.[314] + + 1. indawi. + 2. inawi. + 3. inyuhu. + 4. inkunowi. + 5. inkutaa. + 6. inda-towi = 1 + 5. + 7. ine-towi = 2 + 5. + 8. ine-ukunowi = 2-4. + 9. imuratadahata = 10 - 1? + 10. inda-hata. + + + CORA.[315] + + 1. ceaut. + 2. huapoa. + 3. huaeica. + 4. moacua. + 5. anxuvi. + 6. a-cevi = [5] + 1. + 7. a-huapoa = [5] + 2. + 8. a-huaeica = [5] + 3. + 9. a-moacua = [5] + 4. + 10. tamoamata (akin to moamati, "hand"). + + + AYMARA.[316] + + 1. maya. + 2. paya. + 3. kimsa. + 4. pusi. + 5. piska. + 6. tsokta. + 7. pa-kalko = 2 + 5. + 8. kimsa-kalko = 3 + 5. + 9. pusi-kalko = 4 + 5. + 10. tunka. + + + CARIBS OF ESSEQUIBO, GUIANA.[317] + + 1. oween. + 2. oko. + 3. oroowa. + 4. oko-baimema. + 5. wineetanee = 1 hand. + 6. owee-puimapo = 1 again? + 7. oko-puimapo = 2 again? + 8. oroowa-puimapo = 3 again? + 9. oko-baimema-puimapo = 4 again? + 10. oween-abatoro. + + + CARIB.[318] (ROUCOUYENNE?) + + 1. aban, amoin. + 2. biama. + 3. eleoua. + 4. biam-bouri = 2 again? + 5. ouacabo-apourcou-aban-tibateli. + 6. aban laoyagone-ouacabo-apourcou. + 7. biama laoyagone-ouacabo-apourcou. + 8. eleoua laoyagone-ouacabo-apourcou. + 9. ---- + 10. chon noucabo. + +It is unfortunate that the meanings of these remarkable numerals cannot be +given. The counting is evidently quinary, but the terms used must have been +purely descriptive expressions, having their origin undoubtedly in certain +gestures or finger motions. The numerals obtained from this region, and +from the tribes to the south and east of the Carib country, are especially +rich in digital terms, and an analysis of the above numerals would probably +show clearly the mental steps through which this people passed in +constructing the rude scale which served for the expression of their ideas +of number. + + + KIRIRI.[319] + + 1. biche. + 2. watsani. + 3. watsani dikie. + 4. sumara oroba. + 5. mi biche misa = 1 hand. + 6. mirepri bu-biche misa sai. + 7. mirepri watsani misa sai. + 8. mirepri watsandikie misa sai. + 9. mirepri sumara oraba sai. + 10. mikriba misa sai = both hands. + + + CAYUBABA[320] + + 1. pebi. + 2. mbeta. + 3. kimisa. + 4. pusi. + 5. pisika. + 6. sukuta. + 7. pa-kaluku = 2 again? + 8. kimisa-kaluku = 3 again? + 9. pusu-kaluku = 4 again? + 10. tunka. + + + SAPIBOCONA[320] + + 1. karata. + 2. mitia. + 3. kurapa. + 4. tsada. + 5. maidara (from _arue_, hand). + 6. karata-rirobo = 1 hand with. + 7. mitia-rirobo = 2 hand with. + 8. kurapa-rirobo = 3 hand with. + 9. tsada-rirobo = 4 hand with. + 10. bururutse = hand hand. + + + TICUNA.[321] + + 1. hueih. + 2. tarepueh. + 3. tomepueh. + 4. aguemoujih + 5. hueamepueh. + 6. naimehueapueh = 5 + 1. + 7. naimehueatareh = 5 + 2. + 8. naimehueatameapueh = 5 + 3. + 9. gomeapueh = 10 - 1. + 10. gomeh. + + + YANUA.[322] + + 1. tckini. + 2. nanojui. + 3. munua. + 4. nairojuino = 2d 2. + 5. tenaja. + 6. teki-natea = 1 again? + 7. nanojui-natea = 2 again? + 8. munua-natea = 3 again? + 9. nairojuino-natea = 4 again? + 10. huijejuino = 2 x 5? + +The foregoing examples will show with considerable fulness the wide +dispersion of the quinary scale. Every part of the world contributes its +share except Europe, where the only exceptions to the universal use of the +decimal system are the half-dozen languages, which still linger on its +confines, whose number base is the vigesimal. Not only is there no living +European tongue possessing a quinary number system, but no trace of this +method of counting is found in any of the numerals of the earlier forms of +speech, which have now become obsolete. The only possible exceptions of +which I can think are the Greek [Greek: pempazein], to count by fives, and +a few kindred words which certainly do hint at a remote antiquity in which +the ancestors of the Greeks counted on their fingers, and so grouped their +units into fives. The Roman notation, the familiar I., II., III., IV. +(originally IIII.), V., VI., etc., with equal certainty suggests quinary +counting, but the Latin language contains no vestige of anything of the +kind, and the whole range of Latin literature is silent on this point, +though it contains numerous references to finger counting. It is quite +within the bounds of possibility that the prehistoric nations of Europe +possessed and used a quinary numeration. But of these races the modern +world knows nothing save the few scanty facts that can be gathered from the +stone implements which have now and then been brought to light. Their +languages have perished as utterly as have the races themselves, and +speculation concerning them is useless. Whatever their form of numeration +may have been, it has left no perceptible trace on the languages by which +they were succeeded. Even the languages of northern and central Europe +which were contemporary with the Greek and Latin of classical times have, +with the exception of the Celtic tongues of the extreme North-west, left +behind them but meagre traces for the modern student to work on. We presume +that the ancient Gauls and Goths, Huns and Scythians, and other barbarian +tribes had the same method of numeration that their descendants now have; +and it is a matter of certainty that the decimal scale was, at that time, +not used with the universality which now obtains; but wherever the decimal +was not used, the universal method was vigesimal; and that the quinary ever +had anything of a foothold in Europe is only to be guessed from its +presence to-day in almost all of the other corners of the world. + +From the fact that the quinary is that one of the three natural scales with +the smallest base, it has been conjectured that all tribes possess, at some +time in their history, a quinary numeration, which at a later period merges +into either the decimal or the vigesimal, and thus disappears or forms with +one of the latter a mixed system.[323] In support of this theory it is +urged that extensive regions which now show nothing but decimal counting +were, beyond all reasonable doubt, quinary. It is well known, for example, +that the decimal system of the Malays has spread over almost the entire +Polynesian region, displacing whatever native scales it encountered. The +same phenomenon has been observed in Africa, where the Arab traders have +disseminated their own numeral system very widely, the native tribes +adopting it or modifying their own scales in such a manner that the Arab +influence is detected without difficulty. + +In view of these facts, and of the extreme readiness with which a tribe +would through its finger counting fall into the use of the quinary method, +it does not at first seem improbable that the quinary was _the_ original +system. But an extended study of the methods of counting in vogue among the +uncivilized races of all parts of the world has shown that this theory is +entirely untenable. The decimal scale is no less simple in its structure +than the quinary; and the savage, as he extends the limit of his scale from +5 to 6, may call his new number 5-1, or, with equal probability, give it an +entirely new name, independent in all respects of any that have preceded +it. With the use of this new name there may be associated the conception of +"5 and 1 more"; but in such multitudes of instances the words employed show +no trace of any such meaning, that it is impossible for any one to draw, +with any degree of safety, the inference that the signification was +originally there, but that the changes of time had wrought changes in +verbal form so great as to bury it past the power of recovery. A full +discussion of this question need not be entered upon here. But it will be +of interest to notice two or three numeral scales in which the quinary +influence is so faint as to be hardly discernible. They are found in +considerable numbers among the North American Indian languages, as may be +seen by consulting the vocabularies that have been prepared and published +during the last half century.[324] From these I have selected the +following, which are sufficient to illustrate the point in question: + + + QUAPPA. + + 1. milchtih. + 2. nonnepah. + 3. dahghenih. + 4. tuah. + 5. sattou. + 6. schappeh. + 7. pennapah. + 8. pehdaghenih. + 9. schunkkah. + 10. gedeh bonah. + + + TERRABA.[325] + + 1. krara. + 2. krowue. + 3. krom miah. + 4. krob king. + 5. krasch kingde. + 6. terdeh. + 7. kogodeh. + 8. kwongdeh. + 9. schkawdeh. + 10. dwowdeh. + + + MOHICAN + + 1. ngwitloh. + 2. neesoh. + 3. noghhoh. + 4. nauwoh. + 5. nunon. + 6. ngwittus. + 7. tupouwus. + 8. ghusooh. + 9. nauneeweh. + 10. mtannit. + +In the Quappa scale 7 and 8 appear to be derived from 2 and 3, while 6 and +9 show no visible trace of kinship with 1 and 4. In Mohican, on the other +hand, 6 and 9 seem to be derived from 1 and 4, while 7 and 8 have little or +no claim to relationship with 2 and 3. In some scales a single word only is +found in the second quinate to indicate that 5 was originally the base on +which the system rested. It is hardly to be doubted, even, that change +might affect each and every one of the numerals from 5 to 10 or 6 to 9, so +that a dependence which might once have been easily detected is now +unrecognizable. + +But if this is so, the natural and inevitable question follows--might not +this have been the history of all numeral scales now purely decimal? May +not the changes of time have altered the compounds which were once a clear +indication of quinary counting, until no trace remains by which they can be +followed back to their true origin? Perhaps so. It is not in the least +degree probable, but its possibility may, of course, be admitted. But even +then the universality of quinary counting for primitive peoples is by no +means established. In Chapter II, examples were given of races which had no +number base. Later on it was observed that in Australia and South America +many tribes used 2 as their number base; in some cases counting on past 5 +without showing any tendency to use that as a new unit. Again, through the +habit of counting upon the finger joints, instead of the fingers +themselves, the use of 3 as a base is brought into prominence, and 6 and 9 +become 2 threes and 3 threes, respectively, instead of 5 + 1 and 5 + 4. The +same may be noticed of 4. Counting by means of his fingers, without +including the thumbs, the savage begins by dividing into fours instead of +fives. Traces of this form of counting are somewhat numerous, especially +among the North American aboriginal tribes. Hence the quinary form of +counting, however widespread its use may be shown to be, can in no way be +claimed as the universal method of any stage of development in the history +of mankind. + +In the vast majority of cases, the passage from the base to the next +succeeding number in any scale, is clearly defined. But among races whose +intelligence is of a low order, or--if it be permissible to express it in +this way--among races whose number sense is feeble, progression from one +number to the next is not always in accordance with any well-defined law. +After one or two distinct numerals the count may, as in the case of the +Veddas and the Andamans, proceed by finger pantomime and by the repetition +of the same word. Occasionally the same word is used for two successive +numbers, some gesture undoubtedly serving to distinguish the one from the +other in the savage's mind. Examples of this are not infrequent among the +forest tribes of South America. In the Tariana dialect 9 and 10 are +expressed by the same word, _paihipawalianuda;_ in Cobeu, 8 and 9 by +_pepelicoloblicouilini;_ in Barre, 4, 5, and 9 by _ualibucubi._[326] In +other languages the change from one numeral to the next is so slight that +one instinctively concludes that the savage is forming in his own mind +another, to him new, numeral immediately from the last. In such cases the +entire number system is scanty, and the creeping hesitancy with which +progress is made is visible in the forms which the numerals are made to +take. A single illustration or two of this must suffice; but the ones +chosen are not isolated cases. The scale of the Macunis,[327] one of the +numerous tribes of Brazil, is + + 1. pocchaenang. + 2. haihg. + 3. haigunhgnill. + 4. haihgtschating. + 5. haihgtschihating = another 4? + 6. hathig-stchihathing = 2-4? + 7. hathink-tschihathing = 2-5? + 8. hathink-tschihating = 2 x 4? + +The complete absence of--one is tempted to say--any rhyme or reason from +this scale is more than enough to refute any argument which might tend to +show that the quinary, or any other scale, was ever the sole number scale +of primitive man. Irregular as this is, the system of the Montagnais fully +matches it, as the subjoined numerals show:[328] + + 1. inl'are. + 2. nak'e. + 3. t'are. + 4. dinri. + 5. se-sunlare. + 6. elkke-t'are = 2 x 3. + 7. t'a-ye-oyertan = 10 - 3, + or inl'as dinri = 4 + 3? + 8. elkke-dinri = 2 x 4. + 9. inl'a-ye-oyertan = 10 - 1. + 10. onernan. + + + + + +CHAPTER VII. + +THE VIGESIMAL SYSTEM. + + +In its ordinary development the quinary system is almost sure to merge into +either the decimal or the vigesimal system, and to form, with one or the +other or both of these, a mixed system of counting. In Africa, Oceanica, +and parts of North America, the union is almost always with the decimal +scale; while in other parts of the world the quinary and the vigesimal +systems have shown a decided affinity for each other. It is not to be +understood that any geographical law of distribution has ever been observed +which governs this, but merely that certain families of races have shown a +preference for the one or the other method of counting. These families, +disseminating their characteristics through their various branches, have +produced certain groups of races which exhibit a well-marked tendency, here +toward the decimal, and there toward the vigesimal form of numeration. As +far as can be ascertained, the choice of the one or the other scale is +determined by no external circumstances, but depends solely on the mental +characteristics of the tribes themselves. Environment does not exert any +appreciable influence either. Both decimal and vigesimal numeration are +found indifferently in warm and in cold countries; in fruitful and in +barren lands; in maritime and in inland regions; and among highly civilized +or deeply degraded peoples. + +Whether or not the principal number base of any tribe is to be 20 seems to +depend entirely upon a single consideration; are the fingers alone used as +an aid to counting, or are both fingers and toes used? If only the fingers +are employed, the resulting scale must become decimal if sufficiently +extended. If use is made of the toes in addition to the fingers, the +outcome must inevitably be a vigesimal system. Subordinate to either one of +these the quinary may and often does appear. It is never the principal base +in any extended system. + +To the statement just made respecting the origin of vigesimal counting, +exception may, of course, be taken. In the case of numeral scales like the +Welsh, the Nahuatl, and many others where the exact meanings of the +numerals cannot be ascertained, no proof exists that the ancestors of these +peoples ever used either finger or toe counting; and the sweeping statement +that any vigesimal scale is the outgrowth of the use of these natural +counters is not susceptible of proof. But so many examples are met with in +which the origin is clearly of this nature, that no hesitation is felt in +putting the above forward as a general explanation for the existence of +this kind of counting. Any other origin is difficult to reconcile with +observed facts, and still more difficult to reconcile with any rational +theory of number system development. Dismissing from consideration the +quinary scale, let us briefly examine once more the natural process of +evolution through which the decimal and the vigesimal scales come into +being. After the completion of one count of the fingers the savage +announces his result in some form which definitely states to his mind the +fact that the end of a well-marked series has been reached. Beginning +again, he now repeats his count of 10, either on his own fingers or on the +fingers of another. With the completion of the second 10 the result is +announced, not in a new unit, but by means of a duplication of the term +already used. It is scarcely credible that the unit unconsciously adopted +at the termination of the first count should now be dropped, and a new one +substituted in its place. When the method here described is employed, 20 is +not a natural unit to which higher numbers may be referred. It is wholly +artificial; and it would be most surprising if it were adopted. But if the +count of the second 10 is made on the toes in place of the fingers, the +element of repetition which entered into the previous method is now +wanting. Instead of referring each new number to the 10 already completed, +the savage is still feeling his way along, designating his new terms by +such phrases as "1 on the foot," "2 on the other foot," etc. And now, when +20 is reached, a single series is finished instead of a double series as +before; and the result is expressed in one of the many methods already +noticed--"one man," "hands and feet," "the feet finished," "all the fingers +of hands and feet," or some equivalent formula. Ten is no longer the +natural base. The number from which the new start is made is 20, and the +resulting scale is inevitably vigesimal. If pebbles or sticks are used +instead of fingers, the system will probably be decimal. But back of the +stick and pebble counting the 10 natural counters always exist, and to them +we must always look for the origin of this scale. + +In any collection of the principal vigesimal number systems of the world, +one would naturally begin with those possessed by the Celtic races of +Europe. These races, the earliest European peoples of whom we have any +exact knowledge, show a preference for counting by twenties, which is +almost as decided as that manifested by Teutonic races for counting by +tens. It has been conjectured by some writers that the explanation for this +was to be found in the ancient commercial intercourse which existed between +the Britons and the Carthaginians and Phoenicians, whose number systems +showed traces of a vigesimal tendency. Considering the fact that the use of +vigesimal counting was universal among Celtic races, this explanation is +quite gratuitous. The reason why the Celts used this method is entirely +unknown, and need not concern investigators in the least. But the fact that +they did use it is important, and commands attention. The five Celtic +languages, Breton, Irish, Welsh, Manx, and Gaelic, contain the following +well-defined vigesimal scales. Only the principal or characteristic +numerals are given, those being sufficient to enable the reader to follow +intelligently the growth of the systems. Each contains the decimal element +also, and is, therefore, to be regarded as a mixed decimal-vigesimal +system. + + + IRISH.[329] + + 10. deic. + 20. fice. + 30. triocad = 3-10 + 40. da ficid = 2-20. + 50. caogad = 5-10. + 60. tri ficid = 3-20. + 70. reactmoga = 7-10. + 80. ceitqe ficid = 4-20. + 90. nocad = 9-10. + 100. cead. + 1000. mile. + + + GAELIC.[330] + + 10. deich. + 20. fichead. + 30. deich ar fichead = 10 + 20. + 40. da fhichead = 2-20. + 50. da fhichead is deich = 40 + 10. + 60. tri fichead = 3-20. + 70. tri fichead is deich = 60 + 10. + 80. ceithir fichead = 4-20. + 90. ceithir fichead is deich = 80 + 10. + 100. ceud. + 1000. mile. + + + WELSH.[331] + + 10. deg. + 20. ugain. + 30. deg ar hugain = 10 + 20. + 40. deugain = 2-20. + 50. deg a deugain = 10 + 40. + 60. trigain = 3-20. + 70. deg a thrigain = 10 + 60. + 80. pedwar ugain = 4-20. + 90. deg a pedwar ugain = 80 + 10. + 100. cant. + + + MANX.[332] + + 10. jeih. + 20. feed. + 30. yn jeih as feed = 10 + 20. + 40. daeed = 2-20. + 50. jeih as daeed = 10 + 40. + 60. three-feed = 3-20. + 70. three-feed as jeih = 60 + 10. + 80. kiare-feed = 4-20. + 100. keead. + 1000. thousane, or jeih cheead. + + + BRETON.[333] + + 10. dec. + 20. ueguend. + 30. tregond = 3-10. + 40. deu ueguend = 2-20. + 50. hanter hand = half hundred. + 60. tri ueguend = 3-20. + 70. dec ha tri ueguend = 10 + 60. + 80. piar ueguend = 4-20. + 90. dec ha piar ueguend = 10 + 80. + 100. cand. + 120. hueh ueguend = 6-20. + 140. seih ueguend = 7-20. + 160. eih ueguend = 8-20. + 180. nau ueguend = 9-20. + 200. deu gand = 2-100. + 240. deuzec ueguend = 12-20. + 280. piarzec ueguend = 14-20. + 300. tri hand, or pembzec ueguend. + 400. piar hand = 4-100. + 1000. mil. + +These lists show that the native development of the Celtic number systems, +originally showing a strong preference for the vigesimal method of +progression, has been greatly modified by intercourse with Teutonic and +Latin races. The higher numerals in all these languages, and in Irish many +of the lower also, are seen at a glance to be decimal. Among the scales +here given the Breton, the legitimate descendant of the ancient Gallic, is +especially interesting; but here, just as in the other Celtic tongues, when +we reach 1000, the familiar Latin term for that number appears in the +various corruptions of _mille_, 1000, which was carried into the Celtic +countries by missionary and military influences. + +In connection with the Celtic language, mention must be made of the +persistent vigesimal element which has held its place in French. The +ancient Gauls, while adopting the language of their conquerors, so far +modified the decimal system of Latin as to replace the natural _septante_, +70, _octante_, 80, _nonante_, 90, by _soixante-dix_, 60-10, _quatre-vingt_, +4-20, and _quatrevingt-dix_, 4-20-10. From 61 to 99 the French method of +counting is wholly vigesimal, except for the presence of the one word +_soixante_. In old French this element was still more pronounced. +_Soixante_ had not yet appeared; and 60 and 70 were _treis vinz_, 3-20, and +_treis vinz et dis_, 3-20 and 10 respectively. Also, 120 was _six vinz_, +6-20, 140 was _sept-vinz_, etc.[334] How far this method ever extended in +the French language proper, it is, perhaps, impossible to say; but from the +name of an almshouse, _les quinze-vingts_,[335] which formerly existed in +Paris, and was designed as a home for 300 blind persons, and from the +_pembzek-ueguent_, 15-20, of the Breton, which still survives, we may infer +that it was far enough to make it the current system of common life. + +Europe yields one other example of vigesimal counting, in the number system +of the Basques. Like most of the Celtic scales, the Basque seems to become +decimal above 100. It does not appear to be related to any other European +system, but to be quite isolated philologically. The higher units, as +_mila_, 1000, are probably borrowed, and not native. The tens in the Basque +scale are:[336] + + 10. hamar. + 20. hogei. + 30. hogei eta hamar = 20 + 10. + 40. berrogei = 2-20. + 50. berrogei eta hamar = 2-20 + 10. + 60. hirurogei = 3-20. + 70. hirurogei eta hamar = 3-20 + 10. + 80. laurogei = 4-20. + 90. laurogei eta hamar = 4-20 + 10. + 100. ehun. + 1000. _milla_. + +Besides these we find two or three numeral scales in Europe which contain +distinct traces of vigesimal counting, though the scales are, as a whole, +decidedly decimal. The Danish, one of the essentially Germanic languages, +contains the following numerals: + + 30. tredive = 3-10. + 40. fyrretyve = 4-10. + 50. halvtredsindstyve = half (of 20) from 3-20. + 60. tresindstyve = 3-20. + 70. halvfierdsindstyve = half from 4-20. + 80. fiirsindstyve = 4-20. + 90. halvfemsindstyve = half from 5-20. + 100. hundrede. + +Germanic number systems are, as a rule, pure decimal systems; and the +Danish exception is quite remarkable. We have, to be sure, such expressions +in English as _three score_, _four score_, etc., and the Swedish, +Icelandic, and other languages of this group have similar terms. Still, +these are not pure numerals, but auxiliary words rather, which belong to +the same category as _pair_, _dozen_, _dizaine_, etc., while the Danish +words just given are the ordinary numerals which form a part of the +every-day vocabulary of that language. The method by which this scale +expresses 50, 70, and 90 is especially noticeable. It will be met with +again, and further examples of its occurrence given. + +In Albania there exists one single fragment of vigesimal numeration, which +is probably an accidental compound rather than the remnant of a former +vigesimal number system. With this single exception the Albanian scale is +of regular decimal formation. A few of the numerals are given for the sake +of comparison:[337] + + 30. tridgiete = 3-10. + 40. dizet = 2-20. + 50. pesedgiete = 5-10. + 60. giastedgiete = 6-10, etc. + +Among the almost countless dialects of Africa we find a comparatively small +number of vigesimal number systems. The powers of the negro tribes are not +strongly developed in counting, and wherever their numeral scales have been +taken down by explorers they have almost always been found to be decimal or +quinary-decimal. The small number I have been able to collect are here +given. They are somewhat fragmentary, but are as complete as it was +possible to make them. + + + AFFADEH.[338] + + 10. dekang. + 20. degumm. + 30. piaske. + 40. tikkumgassih = 20 x 2. + 50. tikkumgassigokang = 20 x 2 + 10. + 60. tikkumgakro = 20 x 3. + 70. dungokrogokang = 20 x 3 + 10. + 80. dukumgade = 20 x 4. + 90. dukumgadegokang = 20 x 4 + 10. + 100. miah (borrowed from the Arabs). + + + IBO.[339] + + 10. iri. + 20. ogu. + 30. ogu n-iri = 20 + 10, + or iri ato = 10 x 3. + 40. ogu abuo = 20 x 2, + or iri anno = 10 x 4. + 100. ogu ise = 20 x 5. + + + VEI.[340] + + 10. tan. + 20. mo bande = a person finished. + 30. mo bande ako tan = 20 + 10. + 40. mo fera bande = 2 x 20. + 100. mo soru bande = 5 persons finished. + + + YORUBA.[341] + + 10. duup. + 20. ogu. + 30. ogbo. + 40. ogo-dzi = 20 x 2. + 60. ogo-ta = 20 x 3. + 80. ogo-ri = 20 x 4. + 100. ogo-ru = 20 x 5. + 120. ogo-fa = 20 x 6. + 140. ogo-dze = 20 x 7. + 160. ogo-dzo = 20 x 8, etc. + + + EFIK.[342] + + 10. duup. + 20. edip. + 30. edip-ye-duup = 20 + 10. + 40. aba = 20 x 2. + 60. ata = 20 x 3. + 80. anan = 20 x 4. + 100. ikie. + +The Yoruba scale, to which reference has already been made, p. 70, again +shows its peculiar structure, by continuing its vigesimal formation past +100 with no interruption in its method of numeral building. It will be +remembered that none of the European scales showed this persistency, but +passed at that point into decimal numeration. This will often be found to +be the case; but now and then a scale will come to our notice whose +vigesimal structure is continued, without any break, on into the hundreds +and sometimes into the thousands. + + + BONGO.[343] + + 10. kih. + 20. mbaba kotu = 20 x 1. + 40. mbaba gnorr = 20 x 2. + 100. mbaba mui = 20 x 5. + + + MENDE.[344] + + 10. pu. + 20. nu yela gboyongo mai = a man finished. + 30. nu yela gboyongo mahu pu = 20 + 10. + 40. nu fele gboyongo = 2 men finished. + 100. nu lolu gboyongo = 5 men finished. + + + NUPE.[345] + + 10. gu-wo. + 20. esin. + 30. gbonwo. + 40. si-ba = 2 x 20. + 50. arota. + 60. sita = 3 x 20. + 70. adoni. + 80. sini = 4 x 20. + 90. sini be-guwo = 80 + 10. + 100. sisun = 5 x 20. + + + LOGONE.[346] + + 10. chkan. + 20. tkam. + 30. tkam ka chkan = 20 + 10. + 40. tkam ksde = 20 x 2. + 50. tkam ksde ka chkan = 40 + 10. + 60. tkam gachkir = 20 x 3. + 100. mia (from Arabic). + 1000. debu. + + + MUNDO.[347] + + 10. nujorquoi. + 20. tiki bere. + 30. tiki bire nujorquoi = 20 + 10. + 40. tiki borsa = 20 x 2. + 50. tike borsa nujorquoi = 40 + 10. + + + MANDINGO.[348] + + 10. tang. + 20. mulu. + 30. mulu nintang = 20 + 10. + 40. mulu foola = 20 x 2. + 50. mulu foola nintang = 40 + 10. + 60. mulu sabba = 20 x 3. + 70. mulu sabba nintang = 60 + 10. + 80. mulu nani = 20 x 4. + 90. mulu nani nintang = 80 + 10. + 100. kemi. + +This completes the scanty list of African vigesimal number systems that a +patient and somewhat extended search has yielded. It is remarkable that the +number is no greater. Quinary counting is not uncommon in the "Dark +Continent," and there is no apparent reason why vigesimal reckoning should +be any less common than quinary. Any one investigating African modes of +counting with the material at present accessible, will find himself +hampered by the fact that few explorers have collected any except the first +ten numerals. This leaves the formation of higher terms entirely unknown, +and shows nothing beyond the quinary or non-quinary character of the +system. Still, among those which Stanley, Schweinfurth, Salt, and others +have collected, by far the greatest number are decimal. As our knowledge of +African languages is extended, new examples of the vigesimal method may be +brought to light. But our present information leads us to believe that they +will be few in number. + +In Asia the vigesimal system is to be found with greater frequency than in +Europe or Africa, but it is still the exception. As Asiatic languages are +much better known than African, it is probable that the future will add but +little to our stock of knowledge on this point. New instances of counting +by twenties may still be found in northern Siberia, where much ethnological +work yet remains to be done, and where a tendency toward this form of +numeration has been observed to exist. But the total number of Asiatic +vigesimal scales must always remain small--quite insignificant in +comparison with those of decimal formation. + +In the Caucasus region a group of languages is found, in which all but +three or four contain vigesimal systems. These systems are as follows: + + + ABKHASIA.[349] + + 10. zpha-ba. + 20. gphozpha = 2 x 10. + 30. gphozphei zphaba = 20 + 10. + 40. gphin-gphozpha = 2 x 20. + 60. chin-gphozpha = 3 x 20. + 80. phsin-gphozpha = 4 x 20. + 100. sphki. + + + AVARI + + 10. antsh-go. + 20. qo-go. + 30. lebergo. + 40. khi-qogo = 2 x 20. + 50. khiqojalda antshgo = 40 + 10. + 60. lab-qogo = 3 x 20. + 70. labqojalda antshgo = 60 + 10. + 80. un-qogo = 4 x 20. + 100. nusgo. + + + KURI + + 10. tshud. + 20. chad. + 30. channi tshud = 20 + 10. + 40. jachtshur. + 50. jachtshurni tshud = 40 + 10. + 60. put chad = 3 x 20. + 70. putchanni tshud = 60 + 10. + 80. kud-chad = 4 x 20. + 90. kudchanni tshud = 80 + 10. + 100. wis. + + + UDI + + 10. witsh. + 20. qa. + 30. sa-qo-witsh = 20 + 10. + 40. pha-qo = 2 x 20. + 50. pha-qo-witsh = 40 + 10. + 60. chib-qo = 3 x 20. + 70. chib-qo-witsh = 60 + 10. + 80. bip-qo = 4 x 20. + 90. bip-qo-witsh = 80 + 10. + 100. bats. + 1000. hazar (Persian). + + + TCHETCHNIA + + 10. ith. + 20. tqa. + 30. tqe ith = 20 + 10. + 40. sauz-tqa = 2 x 20. + 50. sauz-tqe ith = 40 + 10. + 60. chuz-tqa = 3 x 20. + 70. chuz-tqe ith = 60 + 10. + 80. w-iez-tqa = 4 x 20. + 90. w-iez-tqe ith = 80 + 10. + 100. b'e. + 1000. ezir (akin to Persian). + + + THUSCH + + 10. itt. + 20. tqa. + 30. tqa-itt = 20 + 10. + 40. sauz-tq = 2 x 20. + 50. sauz-tqa-itt = 40 + 10. + 60. chouz-tq = 3 x 20. + 70. chouz-tqa-itt = 60 + 10. + 80. dhewuz-tq = 4 x 20. + 90. dhewuz-tqa-itt = 80 + 10. + 100. phchauz-tq = 5 x 20. + 200. itsha-tq = 10 x 20. + 300. phehiitsha-tq = 15 x 20. + 1000. satsh tqauz-tqa itshatqa = 2 x 20 x 20 + 200. + + + GEORGIA + + 10. athi. + 20. otsi. + 30. ots da athi = 20 + 10. + 40. or-m-otsi = 2 x 20. + 50. ormots da athi = 40 + 10. + 60. sam-otsi = 3 x 20. + 70. samots da athi = 60 + 10. + 80. othch-m-otsi = 4 x 20. + 90. othmots da athi = 80 + 10. + 100. asi. + 1000. ath-asi = 10 x 100. + + + LAZI + + 10. wit. + 20. oets. + 30. oets do wit = 20 x 10. + 40. dzur en oets = 2 x 20. + 50. dzur en oets do wit = 40 + 10. + 60. dzum en oets = 3 x 20. + 70. dzum en oets do wit = 60 + 10. + 80. otch-an-oets = 4 x 20. + 100. os. + 1000. silia (akin to Greek). + + + CHUNSAG.[350] + + 10. ants-go. + 20. chogo. + 30. chogela antsgo = 20 + 10. + 40. kichogo = 2 x 20. + 50. kichelda antsgo = 40 + 10. + 60. taw chago = 3 x 20. + 70. taw chogelda antsgo = 60 + 10. + 80. uch' chogo = 4 x 20. + 90. uch' chogelda antsgo. + 100. nusgo. + 1000. asargo (akin to Persian). + + + DIDO.[351] + + 10. zino. + 20. ku. + 30. kunozino. + 40. kaeno ku = 2 x 20. + 50. kaeno kuno zino = 40 + 10. + 60. sonno ku = 3 x 20. + 70. sonno kuno zino = 60 + 10. + 80. uino ku = 4 x 20. + 90. uino huno zino = 80 + 10. + 100. bischon. + 400. kaeno kuno zino = 40 x 10. + + + AKARI + + 10. entzelgu. + 20. kobbeggu. + 30. lowergu. + 40. kokawu = 2 x 20. + 50. kikaldanske = 40 + 10. + 60. secikagu. + 70. kawalkaldansku = 3 x 20 + 10. + 80. onkuku = 4 x 20. + 90. onkordansku = 4 x 20 + 10. + 100. nosku. + 1000. askergu (from Persian). + + + CIRCASSIA + + 10. psche. + 20. to-tsch. + 30. totsch-era-pschirre = 20 + 10. + 40. ptl'i-sch = 4 x 10. + 50. ptl'isch-era-pschirre = 40 + 10. + 60. chi-tsch = 6 x 10. + 70. chitsch-era-pschirre = 60 + 10. + 80. toshitl = 20 x 4? + 90. toshitl-era-pschirre = 80 + 10. + 100. scheh. + 1000. min (Tartar) or schi-psche = 100 x 10. + +The last of these scales is an unusual combination of decimal and +vigesimal. In the even tens it is quite regularly decimal, unless 80 is of +the structure suggested above. On the other hand, the odd tens are formed +in the ordinary vigesimal manner. The reason for this anomaly is not +obvious. I know of no other number system that presents the same +peculiarity, and cannot give any hypothesis which will satisfactorily +account for its presence here. In nearly all the examples given the decimal +becomes the leading element in the formation of all units above 100, just +as was the case in the Celtic scales already noticed. + +Among the northern tribes of Siberia the numeral scales appear to be ruder +and less simple than those just examined, and the counting to be more +consistently vigesimal than in any scale we have thus far met with. The two +following examples are exceedingly interesting, as being among the best +illustrations of counting by twenties that are to be found anywhere in the +Old World. + + + TSCHUKSCHI.[352] + + 10. migitken = both hands. + 20. chlik-kin = a whole man. + 30. chlikkin mingitkin parol = 20 + 10. + 40. nirach chlikkin = 2 x 20. + 100. milin chlikkin = 5 x 20. + 200. mingit chlikkin = 10 x 20, _i.e._ 10 men. + 1000. miligen chlin-chlikkin = 5 x 200, _i.e._ five (times) 10 men. + + + AINO.[353] + + 10. wambi. + 20. choz. + 30. wambi i-doehoz = 10 from 40. + 40. tochoz = 2 x 20. + 50. wambi i-richoz = 10 from 60. + 60. rechoz = 3 x 20. + 70. wambi [i?] inichoz = 10 from 80. + 80. inichoz = 4 x 20. + 90. wambi aschikinichoz = 10 from 100. + 100. aschikinichoz = 5 x 20. + 110. wambi juwanochoz = 10 from 120. + 120. juwano choz = 6 x 20. + 130. wambi aruwanochoz = 10 from 140. + 140. aruwano choz = 7 x 20. + 150. wambi tubischano choz = 10 from 160. + 160. tubischano choz = 8 x 20. + 170. wambi schnebischano choz = 10 from 180. + 180. schnebischano choz = 9 x 20. + 190. wambi schnewano choz = 10 from 200. + 200. schnewano choz = 10 x 20. + 300. aschikinichoz i gaschima chnewano choz = 5 x 20 + 10 x 20. + 400. toschnewano choz = 2 x (10 x 20). + 500. aschikinichoz i gaschima toschnewano choz = 100 + 400. + 600. reschiniwano choz = 3 x 200. + 700. aschikinichoz i gaschima reschiniwano choz = 100 + 600. + 800. inischiniwano choz = 4 x 200. + 900. aschikinichoz i gaschima inischiniwano choz = 100 + 800. + 1000. aschikini schinewano choz = 5 x 200. + 2000. wanu schinewano choz = 10 x (10 x 20). + +This scale is in one sense wholly vigesimal, and in another way it is not +to be regarded as pure, but as mixed. Below 20 it is quinary, and, however +far it might be extended, this quinary element would remain, making the +scale quinary-vigesimal. But in another sense, also, the Aino system is not +pure. In any unmixed vigesimal scale the word for 400 must be a simple +word, and that number must be taken as the vigesimal unit corresponding to +100 in the decimal scale. But the Ainos have no simple numeral word for any +number above 20, forming all higher numbers by combinations through one or +more of the processes of addition, subtraction, and multiplication. The +only number above 20 which is used as a unit is 200, which is expressed +merely as 10 twenties. Any even number of hundreds, or any number of +thousands, is then indicated as being so many times 10 twenties; and the +odd hundreds are so many times 10 twenties, plus 5 twenties more. This +scale is an excellent example of the cumbersome methods used by uncivilized +races in extending their number systems beyond the ordinary needs of daily +life. + +In Central Asia a single vigesimal scale comes to light in the following +fragment of the Leptscha scale, of the Himalaya region:[354] + + 10. kati. + 40. kafali = 4 x 10, + or kha nat = 2 x 20. + 50. kafano = 5 x 10, + or kha nat sa kati = 2 x 20 + 10. + 100. gjo, or kat. + +Further to the south, among the Dravidian races, the vigesimal element is +also found. The following will suffice to illustrate the number systems of +these dialects, which, as far as the material at hand shows, are different +from each other only in minor particulars: + + + MUNDARI.[355] + + 10. gelea. + 20. mi hisi. + 30. mi hisi gelea = 20 + 10. + 40. bar hisi = 2 x 20. + 60. api hisi = 3 x 20. + 80. upun hisi = 4 x 20. + 100. mone hisi = 5 x 20. + +In the Nicobar Islands of the Indian Ocean a well-developed example of +vigesimal numeration is found. The inhabitants of these islands are so low +in the scale of civilization that a definite numeral system of any kind is +a source of some surprise. Their neighbours, the Andaman Islanders, it will +be remembered, have but two numerals at their command; their intelligence +does not seem in any way inferior to that of the Nicobar tribes, and one is +at a loss to account for the superior development of the number sense in +the case of the latter. The intercourse of the coast tribes with traders +might furnish an explanation of the difficulty were it not for the fact +that the numeration of the inland tribes is quite as well developed as that +of the coast tribes; and as the former never come in contact with traders +and never engage in barter of any kind except in the most limited way, the +conclusion seems inevitable that this is merely one of the phenomena of +mental development among savage races for which we have at present no +adequate explanation. The principal numerals of the inland and of the coast +tribes are:[356] + + + INLAND TRIBES COAST TRIBES + + 10. teya. 10. sham. + 20. heng-inai. 20. heang-inai. + 30. heng-inai-tain 30. heang-inai-tanai + = 20 + 5 (couples). = 20 + 5 (couples). + 40. au-inai = 2 x 20. 40. an-inai = 2 x 20. + 100. tain-inai = 5 x 20. 100. tanai-inai = 5 x 20. + 200. teya-inai = 10 x 20. 200. sham-inai = 10 x 20. + 300. teya-tain-inai 300. heang-tanai-inai + = (10 + 5) x 20. = (10 + 5) 20. + 400. heng-teo. 400. heang-momchiama. + +In no other part of the world is vigesimal counting found so perfectly +developed, and, among native races, so generally preferred, as in North and +South America. In the eastern portions of North America and in the extreme +western portions of South America the decimal or the quinary decimal scale +is in general use. But in the northern regions of North America, in western +Canada and northwestern United States, in Mexico and Central America, and +in the northern and western parts of South America, the unit of counting +among the great majority of the native races was 20. The ethnological +affinities of these races are not yet definitely ascertained; and it is no +part of the scope of this work to enter into any discussion of that +involved question. But either through contact or affinity, this form of +numeration spread in prehistoric times over half or more than half of the +western hemisphere. It was the method employed by the rude Eskimos of the +north and their equally rude kinsmen of Paraguay and eastern Brazil; by the +forest Indians of Oregon and British Columbia, and by their more southern +kinsmen, the wild tribes of the Rio Grande and of the Orinoco. And, most +striking and interesting of all, it was the method upon which were based +the numeral systems of the highly civilized races of Mexico, Yucatan, and +New Granada. Some of the systems obtained from the languages of these +peoples are perfect, extended examples of vigesimal counting, not to be +duplicated in any other quarter of the globe. The ordinary unit was, as +would be expected, "one man," and in numerous languages the words for 20 +and man are identical. But in other cases the original meaning of that +numeral word has been lost; and in others still it has a signification +quite remote from that given above. These meanings will be noticed in +connection with the scales themselves, which are given, roughly speaking, +in their geographical order, beginning with the Eskimo of the far north. +The systems of some of the tribes are as follows: + + + ALASKAN ESKIMOS.[357] + + 10. koleet. + 20. enuenok. + 30. enuenok kolinik = 20 + 10. + 40. malho kepe ak = 2 x 20. + 50. malho-kepe ak-kolmik che pah ak to = 2 x 20 + 10. + 60. pingi shu-kepe ak = 3 x 20. + 100. tale ma-kepe ak = 5 x 20. + 400. enue nok ke pe ak = 20 x 20. + + + TCHIGLIT.[358] + + 10. krolit. + 20. kroleti, or innun = man. + 30. innok krolinik-tchikpalik = man + 2 hands. + 40. innum mallerok = 2 men. + 50. adjigaynarmitoat = as many times 10 as the fingers of the hand. + 60. innumipit = 3 men. + 70. innunmalloeronik arveneloerit = 7 men? + 80. innun pinatcunik arveneloerit = 8 men? + 90. innun tcitamanik arveneloerit = 9 men? + 100. itchangnerkr. + 1000. itchangner-park = great 100. + +The meanings for 70, 80, 90, are not given by Father Petitot, but are of +such a form that the significations seem to be what are given above. Only a +full acquaintance with the Tchiglit language would justify one in giving +definite meanings to these words, or in asserting that an error had been +made in the numerals. But it is so remarkable and anomalous to find the +decimal and vigesimal scales mingled in this manner that one involuntarily +suspects either incompleteness of form, or an actual mistake. + + + TLINGIT.[359] + + 10. djinkat = both hands? + 20. tle ka = 1 man. + 30. natsk djinkat = 3 x 10. + 40. dak'on djinkat = 4 x 10. + 50. kedjin djinkat = 5 x 10. + 60. tle durcu djinkat = 6 x 10. + 70. daqa durcu djinkat = 7 x 10. + 80. natska durcu djinkat = 8 x 10. + 90. gocuk durcu djinkat = 9 x 10. + 100. kedjin ka = 5 men, or 5 x 20. + 200. djinkat ka = 10 x 20. + 300. natsk djinkat ka = 30 men. + 400. dak'on djinkat ka = 40 men. + +This scale contains a strange commingling of decimal and vigesimal +counting. The words for 20, 100, and 200 are clear evidence of vigesimal, +while 30 to 90, and the remaining hundreds, are equally unmistakable proof +of decimal, numeration. The word _ka_, man, seems to mean either 10 or 20; +a most unusual occurrence. The fact that a number system is partly decimal +and partly vigesimal is found to be of such frequent occurrence that this +point in the Tlingit scale need excite no special wonder. But it is +remarkable that the same word should enter into numeral composition under +such different meanings. + + + NOOTKA.[360] + + 10. haiu. + 20. tsakeits. + 30. tsakeits ic haiu = 20 + 10. + 40. atlek = 2 x 20. + 60. katstsek = 3 x 20. + 80. moyek = 4 x 20. + 100. sutc'ek = 5 x 20. + 120. nop'ok = 6 x 20. + 140. atlpok = 7 x 20. + 160. atlakutlek = 8 x 20. + 180. ts'owakutlek = 9 x 20. + 200. haiuk = 10 x 20. + +This scale is quinary-vigesimal, with no apparent decimal element in its +composition. But the derivation of some of the terms used is detected with +difficulty. In the following scale the vigesimal structure is still more +obscure. + + + TSIMSHIAN.[361] + + 10. gy'ap. + 20. kyedeel = 1 man. + 30. gulewulgy'ap. + 40. t'epqadalgyitk, or tqalpqwulgyap. + 50. kctoncwulgyap. + 100. kcenecal. + 200. k'pal. + 300. k'pal te kcenecal = 200 + 100. + 400. kyedal. + 500. kyedal te kcenecal = 400 + 100. + 600. gulalegyitk. + 700. gulalegyitk te kcenecal = 600 + 100. + 800. tqalpqtalegyitk. + 900. tqalpqtalegyitk te kcenecal = 800 + 100. + 1000. k'pal. + +To the unobservant eye this scale would certainly appear to contain no more +than a trace of the vigesimal in its structure. But Dr. Boas, who is one of +the most careful and accurate of investigators, says in his comment on this +system: "It will be seen at once that this system is quinary-vigesimal.... +In 20 we find the word _gyat_, man. The hundreds are identical with the +numerals used in counting men (see p. 87), and then the quinary-vigesimal +system is most evident." + + + RIO NORTE INDIANS.[362] + + 20. taiguaco. + 30. taiguaco co juyopamauj ajte = 20 + 2 x 5. + 40. taiguaco ajte = 20 x 2. + 50. taiguaco ajte co juyopamauj ajte = 20 x 2 + 5 x 2. + + + CARIBS OF ESSIQUIBO, GUIANA + + 10. oween-abatoro. + 20. owee-carena = 1 person. + 40. oko-carena = 2 persons. + 60. oroowa-carena = 3 persons. + + + OTOMI + + 10. ra-tta. + 20. na-te. + 30. na-te-m'a-ratta = 20 + 10. + 40. yo-te = 2 x 30. + 50. yote-m'a-ratta = 2 x 20 + 10. + 60. hiu-te = 3 x 20. + 70. hiute-m'a-ratta = 3 x 20 + 10. + 80. gooho-rate = 4 x 20. + 90. gooho-rate-m'a ratta = 4 x 20 + 10. + 100. cytta-te = 5 x 20, + or nanthebe = 1 x 100. + + + MAYA, YUCATAN.[363] + + 1. hun. + 10. lahun = it is finished. + 20. hunkal = a measure, or more correctly, a fastening together. + 30. lahucakal = 40 - 10? + 40. cakal = 2 x 20. + 50. lahuyoxkal = 60 - 10. + 60. oxkal = 3 x 20. + 70. lahucankal = 80 - 10. + 80. cankal = 4 x 20. + 90. lahuyokal = 100 - 10. + 100. hokal = 5 x 20. + 110. lahu uackal = 120 - 10. + 120. uackal = 6 x 20. + 130. lahu uuckal = 140 - 10. + 140. uuckal = 7 x 20. + 200. lahuncal = 10 x 20. + 300. holhukal = 15 x 20. + 400. hunbak = 1 tying around. + 500. hotubak. + 600. lahutubak + 800. calbak = 2 x 400. + 900. hotu yoxbak. + 1000. lahuyoxbak. + 1200. oxbak = 3 x 400. + 2000. capic (modern). + 8000. hunpic = 1 sack. + 16,000. ca pic (ancient). + 160,000. calab = a filling full + 3,200,000. kinchil. + 64,000,000. hunalau. + +In the Maya scale we have one of the best and most extended examples of +vigesimal numeration ever developed by any race. To show in a more striking +and forcible manner the perfect regularity of the system, the following +tabulation is made of the various Maya units, which will correspond to the +"10 units make one ten, 10 tens make one hundred, 10 hundreds make one +thousand," etc., which old-fashioned arithmetic compelled us to learn in +childhood. The scale is just as regular by twenties in Maya as by tens in +English. It is[364] + + 20 hun = 1 kal = 20. + 20 kal = 1 bak = 400. + 20 bak = 1 pic = 8000. + 20 pic = 1 calab = 160,000. + 20 calab = 1 { kinchil } = 3,200,000. + { tzotzceh } + 20 kinchil = 1 alau = 64,000,000. + +The original meaning of _pic_, given in the scale as "a sack," was rather +"a short petticoat, somtimes used as a sack." The word _tzotzceh_ signified +"deerskin." No reason can be given for the choice of this word as a +numeral, though the appropriateness of the others is sufficiently manifest. +No evidence of digital numeration appears in the first 10 units, but, +judging from the almost universal practice of the Indian tribes of both +North and South America, such may readily have been the origin of Maya +counting. Whatever its origin, it certainly expanded and grew into a system +whose perfection challenges our admiration. It was worthy of the splendid +civilization of this unfortunate race, and, through its simplicity and +regularity, bears ample testimony to the intellectual capacity which +originated it. + +The only example of vigesimal reckoning which is comparable with that of +the Mayas is the system employed by their northern neighbours, the Nahuatl, +or, as they are more commonly designated, the Aztecs of Mexico. This system +is quite as pure and quite as simple as the Maya, but differs from it in +some important particulars. In its first 20 numerals it is quinary (see p. +141), and as a system must be regarded as quinary-vigesimal. The Maya scale +is decimal through its first 20 numerals, and, if it is to be regarded as a +mixed scale, must be characterized as decimal-vigesimal. But in both these +instances the vigesimal element preponderates so strongly that these, in +common with their kindred number systems of Mexico, Yucatan, and Central +America, are always thought of and alluded to as vigesimal scales. On +account of its importance, the Nahuatl system[365] is given in fuller +detail than most of the other systems I have made use of. + + 10. matlactli = 2 hands. + 20. cempoalli = 1 counting. + 21. cempoalli once = 20-1. + 22. cempoalli omome = 20-2. + 30. cempoalli ommatlactli = 20-10. + 31. cempoalli ommatlactli once = 20-10-1. + 40. ompoalli = 2 x 20. + 50. ompoalli ommatlactli = 40-10. + 60. eipoalli, or epoalli, = 3 x 20. + 70. epoalli ommatlactli = 60-10. + 80. nauhpoalli = 4 x 20. + 90. nauhpoalli ommatlactli = 90-10. + 100. macuilpoalli = 5 x 20. + 120. chiquacempoalli = 6 x 20. + 140. chicompoalli = 7 x 20. + 160. chicuepoalli = 8 x 20. + 180. chiconauhpoalli = 9 x 20. + 200. matlacpoalli = 10 x 20. + 220. matlactli oncempoalli = 11 x 20. + 240. matlactli omompoalli = 12 x 20. + 260. matlactli omeipoalli = 13 x 20. + 280. matlactli onnauhpoalli = 14 x 20. + 300. caxtolpoalli = 15 x 20. + 320. caxtolli oncempoalli. + 399. caxtolli onnauhpoalli ipan caxtolli onnaui = 19 x 20 + 19. + 400. centzontli = 1 bunch of grass, or 1 tuft of hair. + 800. ometzontli = 2 x 400. + 1200. eitzontli = 3 x 400. + 7600. caxtolli onnauhtzontli = 19 x 400. + 8000. cenxiquipilli, or cexiquipilli. + 160,000. cempoalxiquipilli = 20 x 8000. + 3,200,000. centzonxiquipilli = 400 x 8000. + 64,000,000. cempoaltzonxiquipilli = 20 x 400 x 8000. + +Up to 160,000 the Nahuatl system is as simple and regular in its +construction as the English. But at this point it fails in the formation of +a new unit, or rather in the expression of its new unit by a simple word; +and in the expression of all higher numbers it is forced to resort in some +measure to compound terms, just as the English might have done had it not +been able to borrow from the Italian. The higher numeral terms, under such +conditions, rapidly become complex and cumbersome, as the following +analysis of the number 1,279,999,999 shows.[366] The analysis will be +readily understood when it is remembered that _ipan_ signifies plus. +_Caxtolli onnauhpoaltzonxiquipilli ipan caxtolli onnauhtzonxiquipilli ipan +caxtolli onnauhpoalxiquipilli ipan caxtolli onnauhxiquipilli ipan caxtolli +onnauhtzontli ipan caxtolli onnauhpoalli ipan caxtolli onnaui;_ _i.e._ +1,216,000,000 + 60,800,000 + 3,040,000 + 152,000 + 7600 + 380 + 19. To +show the compounding which takes place in the higher numerals, the analysis +may be made more literally, thus: + (15 + 4) x 400 x 800 + (15 + 4) x 20 x +8000 + (15 + 4) x 8000 + (15 + 4) x 400 + (15 + 4) x 20 + 15 + 4. Of +course this resolution suffers from the fact that it is given in digits +arranged in accordance with decimal notation, while the Nahuatl numerals +express values by a base twice as great. This gives the effect of a +complexity and awkwardness greater than really existed in the actual use of +the scale. Except for the presence of the quinary element the number just +given is really expressed with just as great simplicity as it could be in +English words if our words "million" and "billion" were replaced by +"thousand thousand" and "thousand thousand thousand." If Mexico had +remained undisturbed by Europeans, and science and commerce had been left +to their natural growth and development, uncompounded words would +undoubtedly have been found for the higher units, 160,000, 3,200,000, etc., +and the system thus rendered as simple as it is possible for a +quinary-vigesimal system to be. + +Other number scales of this region are given as follows: + + + HUASTECA.[367] + + 10. laluh. + 20. hum-inic = 1 man. + 30. hum-inic-lahu = 1 man 10. + 40. tzab-inic = 2 men. + 50. tzab-inic-lahu = 2 men 10. + 60. ox-inic = 3 men. + 70. ox-inic-lahu = 3 men 10. + 80. tze-tnic = 4 men. + 90. tze-ynic-kal-laluh = 4 men and 10. + 100. bo-inic = 5 men. + 200. tzab-bo-inic = 2 x 5 men. + 300. ox-bo-inic = 3 x 5 men. + 400. tsa-bo-inic = 4 x 5 men. + 600. acac-bo-inic = 6 x 5 men. + 800. huaxic-bo-inic = 8 x 5 men. + 1000. xi. + 8000. huaxic-xi = 8-1000. + +The essentially vigesimal character of this system changes in the formation +of some of the higher numerals, and a suspicion of the decimal enters. One +hundred is _boinic_, 5 men; but 200, instead of being simply _lahuh-inic_, +10 men, is _tsa-bo-inic_, 2 x 100, or more strictly, 2 times 5 men. +Similarly, 300 is 3 x 100, 400 is 4 x 100, etc. The word for 1000 is simple +instead of compound, and the thousands appear to be formed wholly on the +decimal base. A comparison of this scale with that of the Nahuatl shows how +much inferior it is to the latter, both in simplicity and consistency. + + + TOTONACO.[368] + + 10. cauh. + 20. puxam. + 30. puxamacauh = 20 + 10. + 40. tipuxam = 2 x 20. + 50. tipuxamacauh = 40 + 10. + 60. totonpuxam = 3 x 20. + 100. quitziz puxum = 5 x 20. + 200. copuxam = 10 x 20. + 400. tontaman. + 1000. titamanacopuxam = 2 x 400 + 200. + +The essential character of the vigesimal element is shown by the last two +numerals. _Tontamen_, the square of 20, is a simple word, and 1000 is, as +it should be, 2 times 400, plus 200. It is most unfortunate that the +numeral for 8000, the cube of 20, is not given. + + + CORA.[369] + + 10. tamoamata. + 20. cei-tevi. + 30. ceitevi apoan tamoamata = 20 + 10. + 40. huapoa-tevi = 2 x 20. + 60. huaeica-tevi = 3 x 20. + 100. anxu-tevi = 5 x 20. + 400. ceitevi-tevi = 20 x 20. + +Closely allied with the Maya numerals and method of counting are those of +the Quiches of Guatemala. The resemblance is so obvious that no detail in +the Quiche scale calls for special mention. + + + QUICHE.[370] + + 10. lahuh. + 20. hu-uinac = 1 man. + 30. hu-uinac-lahuh = 20 + 10. + 40. ca-uinac = 2 men. + 50. lahu-r-ox-kal = -10 + 3 x 20. + 60. ox-kal = 3 x 20. + 70. lahu-u-humuch = -10 + 80. + 80. humuch. + 90. lahu-r-ho-kal = -10 + 100. + 100. hokal. + 1000. o-tuc-rox-o-kal. + +Among South American vigesimal systems, the best known is that of the +Chibchas or Muyscas of the Bogota region, which was obtained at an early +date by the missionaries who laboured among them. This system is much less +extensive than that of some of the more northern races; but it is as +extensive as almost any other South American system with the exception of +the Peruvian, which was, however, a pure decimal system. As has already +been stated, the native races of South America were, as a rule, exceedingly +deficient in regard to the number sense. Their scales are rude, and show +great poverty, both in formation of numeral words and in the actual extent +to which counting was carried. If extended as far as 20, these scales are +likely to become vigesimal, but many stop far short of that limit, and no +inconsiderable number of them fail to reach even 5. In this respect we are +reminded of the Australian scales, which were so rudimentary as really to +preclude any proper use of the word "system" in connection with them. +Counting among the South American tribes was often equally limited, and +even less regular. Following are the significant numerals of the scale in +question: + + + CHIBCHA, OR MUYSCA.[371] + + 10. hubchibica. + 20. quihica ubchihica = thus says the foot, 10 = 10-10, + or gueta = house. + 30. guetas asaqui ubchihica = 20 + 10. + 40. gue-bosa = 20 x 2. + 60. gue-mica = 20 x 3. + 80. gue-muyhica = 20 x 4. + 100. gue-hisca = 20 x 5. + + + NAGRANDA.[372] + + 10. guha. + 20. dino. + 30. 'badinoguhanu = 20 + 10. + 40. apudino = 2 x 20. + 50. apudinoguhanu = 2 x 20 + 10. + 60. asudino = 3 x 20. + 70. asudinoguhanu = 3 x 20 + 10. + 80. acudino = 4 x 20. + 90. acudinoguhanu = 4 x 20 + 10. + 100. huisudino = 5 x 20, + or guhamba = great 10. + 200. guahadino = 10 x 20. + 400. dinoamba = great 20. + 1000. guhaisudino = 10 x 5 x 20. + 2000. hisudinoamba = 5 great 20's. + 4000. guhadinoamba = 10 great 20's. + +In considering the influence on the manners and customs of any people which +could properly be ascribed to the use among them of any other base than 10, +it must not be forgotten that no races, save those using that base, have +ever attained any great degree of civilization, with the exception of the +ancient Aztecs and their immediate neighbours, north and south. For reasons +already pointed out, no highly civilized race has ever used an exclusively +quinary system; and all that can be said of the influence of this mode of +counting is that it gives rise to the habit of collecting objects in groups +of five, rather than of ten, when any attempt is being made to ascertain +their sum. In the case of the subsidiary base 12, for which the Teutonic +races have always shown such a fondness, the dozen and gross of commerce, +the divisions of English money, and of our common weights and measures are +probably an outgrowth of this preference; and the Babylonian base, 60, has +fastened upon the world forever a sexagesimal method of dividing time, and +of measuring the circumference of the circle. + +The advanced civilization attained by the races of Mexico and Central +America render it possible to see some of the effects of vigesimal +counting, just as a single thought will show how our entire lives are +influenced by our habit of counting by tens. Among the Aztecs the universal +unit was 20. A load of cloaks, of dresses, or other articles of convenient +size, was 20. Time was divided into periods of 20 days each. The armies +were numbered by divisions of 8000;[373] and in countless other ways the +vigesimal element of numbers entered into their lives, just as the decimal +enters into ours; and it is to be supposed that they found it as useful and +as convenient for all measuring purposes as we find our own system; as the +tradesman of to-day finds the duodecimal system of commerce; or as the +Babylonians of old found that singularly curious system, the sexagesimal. +Habituation, the laws which the habits and customs of every-day life impose +upon us, are so powerful, that our instinctive readiness to make use of any +concept depends, not on the intrinsic perfection or imperfection which +pertains to it, but on the familiarity with which previous use has invested +it. Hence, while one race may use a decimal, another a quinary-vigesimal, +and another a sexagesimal scale, and while one system may actually be +inherently superior to another, no user of one method of reckoning need +ever think of any other method as possessing practical inconveniences, of +which those employing it are ever conscious. And, to cite a single instance +which illustrates the unconscious daily use of two modes of reckoning in +one scale, we have only to think of the singular vigesimal fragment which +remains to this day imbedded in the numeral scale of the French. In +counting from 70 to 100, or in using any number which lies between those +limits, no Frenchman is conscious of employing a method of numeration less +simple or less convenient in any particular, than when he is at work with +the strictly decimal portions of his scale. He passes from the one style of +counting to the other, and from the second back to the first again, +entirely unconscious of any break or change; entirely unconscious, in fact, +that he is using any particular system, except that which the daily habit +of years has made a part himself. + +Deep regret must be felt by every student of philology, that the primitive +meanings of simple numerals have been so generally lost. But, just as the +pebble on the beach has been worn and rounded by the beating of the waves +and by other pebbles, until no trace of its original form is left, and +until we can say of it now only that it is quartz, or that it is diorite, +so too the numerals of many languages have suffered from the attrition of +the ages, until all semblance of their origin has been lost, and we can say +of them only that they are numerals. Beyond a certain point we can carry +the study neither of number nor of number words. At that point both the +mathematician and the philologist must pause, and leave everything beyond +to the speculations of those who delight in nothing else so much as in pure +theory. + + + + +THE END. + + + + + +INDEX OF AUTHORS. + + +Adam, L., 44, 159, 166, 175. +Armstrong, R.A., 180. +Aymonier, A., 156. + +Bachofen, J.J., 131. +Balbi, A., 151. +Bancroft, H.H., 29, 47, 89, 93, 113, 199. +Barlow, H., 108. +Beauregard, O., 45, 83, 152. +Bellamy, E.W., 9. +Boas, F., 30, 45, 46, 65, 87, 88, 136, 163, 164, 171, 197, 198. +Bonwick, J., 24, 27, 107, 108. +Brinton, D.G., 2, 22, 46, 52, 57, 61, 111, 112, 140, 199, 200. +Burton, R.F., 37, 71. + +Chamberlain, A.F., 45, 65, 93. +Chase, P.E., 99. +Clarke, H., 113. +Codrington, R.H., 16, 95, 96, 136, 138, 145, 153, 154. +Crawfurd, J., 89, 93, 130. +Curr, E.M., 24-27, 104, 107-110, 112. +Cushing, F.H., 13, 48. + +De Flacourt, 8, 9. +De Quincey, T., 35. +Deschamps, M., 28. +Dobrizhoffer, M., 71. +Dorsey, J.O., 59. +Du Chaillu, P.B., 66, 67, 150, 151. +Du Graty, A.M., 138. + +Ellis, A.A., 64, 91. +Ellis, R., 37, 142. +Ellis, W., 83, 119. +Erskine, J.E., 153, 154. + +Flegel, R., 133. + +Gallatin, A., 136, 159, 166, 171, 199, 204, 206, 208. +Galton, F., 4. +Gatschet, A.S., 58, 59, 68. +Gilij, F.S., 54. +Gill, W.W., 18, 118. +Goedel, M., 83, 147. +Grimm, J.L.C., 48. +Groeber, G., 182. +Guillome, J., 181. + +Haddon, A.C., 18, 105. +Hale, H., 61, 65, 93, 114-116, 122, 130, 156, 163, 164, 171. +Hankel, H., 137. +Haumonte, J.D., 44. +Hervas, L., 170. +Humboldt, A. von, 32, 207. +Hyades, M., 22. + +Kelly, J.W., 157, 196. +Kelly, J., 180. +Kleinschmidt, S., 52, 80. + +Lang, J.D., 108. +Lappenberg, J.M., 127. +Latham, R.G., 24, 67, 107. +Leibnitz, G.W. von, 102, 103. +Lloyd, H.E., 7. +Long, C.C., 148, 186. +Long, S.H., 121. +Lubbock, Sir J., 3, 5. +Lull, E.P., 79. + +Macdonald, J., 15. +Mackenzie, A., 26. +Man, E.H., 28, 194. +Mann, A., 47. +Marcoy, P. (Saint Cricq), 23, 168. +Mariner, A., 85. +Martius, C.F. von, 23, 79, 111, 122, 138, 142, 174. +Mason, 112. +Mill, J.S., 3. +Moncelon, M., 142. +Morice, A., 15, 86. +Mueller, Fr., 10, 27, 28, 45, 48, 55, 56, 60, 63, 66, 69, 78, 80, 90, 108, + 111, 121, 122, 130, 136, 139, 146-151, 156-158, 165-167, 185-187, 191, + 193. +Murdoch, J., 30, 49,137. + +Nystron, J.W., 132. + +O'Donovan, J., 180. +Oldfield, A., 29, 77. +Olmos, A. de, 141. + +Parisot, J., 44. +Park, M., 145-147. +Parry, W.E., 32. +Peacock, G., 8, 56, 84, 111, 118, 119, 154, 186. +Petitot, E., 53, 157, 196. +Pott, A.F., 50, 68, 92, 120, 145, 148, 149, 152, 157, 166, 182, 184, 189, + 191, 205. +Pruner-Bey, 10, 104. +Pughe, W.O., 141. + +Ralph, J., 125. +Ray, S.H., 45, 78, 80. +Ridley, W., 57. +Roth, H.L., 79. + +Salt, H., 187. +Sayce, A.H., 75. +Schoolcraft, H.R., 66, 81, 83, 84, 159, 160. +Schroeder, P., 90. +Schweinfurth, G., 143, 146, 149, 186, 187. +Simeon, R., 201. +Spix, J.B. von, 7. +Spurrell, W., 180. +Squier, G.E., 80, 207. +Stanley, H.M., 38, 42, 64, 69, 78, 150, 187. + +Taplin, G., 106. +Thiel, B.A., 172. +Toy, C.H., 70. +Turner, G., 152, 154. +Tylor, E.B., 2, 3, 15, 18, 22, 63, 65, 78, 79, 81, 84, 97, 124. + +Van Eys, J.W., 182. +Vignoli, T., 95. + +Wallace, A.R., 174. +Wells, E.R., jr., 157, 196. +Whewell, W., 3. +Wickersham, J., 96. +Wiener, C., 22. +Williams, W.L., 123. + + + + + +INDEX OF SUBJECTS. + + +Abacus, 19. +Abeokuta, 33. +Abipone, 71, 72. +Abkhasia, 188. +Aboker, 148. +Actuary, Life ins., 19. +Adaize, 162. +Addition, 19, 43, 46, 92. +Adelaide, 108. +Admiralty Islands, 45. +Affadeh, 184. +Africa (African), 9, 16, 28, 29, 32, 33, 38, 42, 47, 64, 66, 69, 78, 80, + 91, 105, 120, 145, 170, 176, 184, 187. +Aino (Ainu), 45, 191. +Akra, 120. +Akari, 190. +Alaska, 157, 196. +Albania, 184. +Albert River, 26. +Aleut, 157. +Algonkin (Algonquin), 45, 92, 161. +Amazon, 23. +Ambrym, 136. +American, 10, 16, 19, 98, 105. +Andaman, 8, 15, 28, 31, 76, 174, 193. +Aneitum, 154. +Animal, 3, 6. +Anthropological, 21. +Apho, 133. +Api, 80, 136, 155. +Apinage, 111. +Arab, 170. +Arawak, 52-54, 135. +Arctic, 29. +Arikara, 46. +Arithmetic, 1, 5, 30, 33, 73, 93. +Aryan, 76, 128-130. +Ashantee, 145. +Asia (Asiatic), 28, 113, 131, 187. +Assiniboine, 66, 92. +Athapaskan,92. +Atlantic, 126. +Aurora, 155. +Australia (Australian), 2, 6, 19, 22, 24-30, 57, 58, 71, 75, 76, 84, 103, + 105, 106, 110, 112, 118, 173, 206. +Avari, 188. +Aymara, 166. +Aztec, 63, 78, 83, 89, 93, 201, 207, 208. + +Babusesse, 38. +Babylonian, 208. +Bagrimma, 148. +Bahnars, 15. +Bakairi, 111. +Balad, 67. +Balenque, 150. +Bambarese, 95. +Banks Islands, 16, 96, 153. +Barea, 151. +Bargaining, 18, 19, 32. +Bari, 136. +Barre, 174. +Basa, 146. +Basque, 40, 182. +Bellacoola, see Bilqula. +Belyando River, 109. +Bengal, Bay of, 28. +Benue, 133. +Betoya, 57, 112, 135, 140. +Bilqula, 46, 164. +Binary, chap. v. +Binin, 149. +Bird-nesting, 5. +Bisaye, 90. +Bogota, 206. +Bolan, 120. +Bolivia, 2, 21. +Bongo, 143, 186. +Bonze, 151. +Bororo, 23. +Botocudo, 22, 31, 48, 71. +Bourke, 108. +Boyne River, 24. +Brazil, 2, 7, 30, 174, 195. +Bretagne (Breton), 120, 129, 181, 182. +British Columbia, 45, 46, 65, 86, 88, 89, 112, 113, 195. +Bullom, 147. +Bunch, 64. +Burnett River, 112. +Bushman, 28, 31. +Butong, 93. + +Caddoe, 162. +Cahuillo, 165. +Calculating machine, 19. +Campa, 22. +Canada, 29, 53, 54, 86, 195. +Canaque, 142, 144. +Caraja, 23. +Carib, 166, 167, 199. +Carnarvon, 35, 36. +Carrier, 86. +Carthaginian, 179. +Caucasus, 188. +Cayriri (see Kiriri), 79. +Cayubaba (Cayubabi), 84, 167. +Celtic, 40, 169, 179, 181, 190. +Cely, Mom, 9. +Central America, 29, 69, 79, 121, 131, 195, 201, 208. +Ceylon, 28. +Chaco, 22. +Champion Bay, 109. +Charles XII., 132. +Cheyenne, 62. +Chibcha, 206. +China (Chinese), 40, 131. +Chippeway, 62, 159, 162. +Chiquito, 2, 6, 21, 40, 71, 76. +Choctaw, 65, 85, 162. +Chunsag, 189. +Circassia, 190. +Cobeu, 174. +Cochin China, 15. +Columbian, 113. +Comanche, 29, 83. +Conibo, 23. +Cooper's Creek, 108. +Cora, 166. +Cotoxo, 111. +Cowrie, 64, 70, 71. +Cree, 91. +Crocker Island, 107. +Crow, 3, 4, 92. +Crusoe, Robinson, 7. +Curetu, 111. + +Dahomey, 71. +Dakota, 81, 91, 92. +Danish, 30, 46, 129, 183. +Darnley Islands, 24. +Delaware, 91, 160. +Demara, 4, 6. +Dene, 86. +Dido, 189. +Dinka, 136, 147. +Dippil, 107. +Division, 19. +Dravidian, 104, 193. +Dual number, 75. +Duluth, 34. +Duodecimal, chap. v. +Dutch, 129. + +Eaw, 24. +Ebon, 152. +Efik, 148, 185. +Encabellada, 22. +Encounter Bay, 108. +Ende, 68, 152. +English, 28, 38-44, 60, 81, 85, 89, 118, 123, 124, 129, 183, 200, 203, 208. +Eromanga, 96, 136, 154. +Eskimo, 16, 30, 31, 32, 36, 48, 51, 52, 54, 61, 64, 83, 137, 157, 159, 195, + 196. +Essequibo, 166. +Europe (European), 27, 39, 168, 169, 179, 182, 183, 185, 204. +Eye, 14, 97. +Eyer's Sand Patch, 26. +Ewe, 64, 91. + +Fall, 163. +Fate, 138, 155. +Fatuhiva, 130. +Feloop, 145. +Fernando Po, 150. +Fiji, 96. +Finger pantomime, 10, 23, 29, 67, 82. +Fingoe, 33. +Fist, 16, 59, 72. +Flinder's River, 24. +Flores, 68, 152. +Forefinger, 12, 15, 16, 54, 61, 91, 113. +Foulah, 147. +Fourth finger, 18. +Frazer's Island, 108. +French, 40, 41, 124, 129, 181, 182, 209. +Fuegan, 22. + +Gaelic, 180. +Galibi, 138. +Gaul, 169, 182. +Georgia, 189. +German, 38-43, 129, 183. +Gesture, 18, 59. +Gola, 151. +Golo, 146. +Gonn Station, 110. +Goth, 169. +Greek, 86, 129, 168, 169. +Green Island, 45. +Greenland, 29, 52, 80, 158. +Guachi, 23, 31. +Guarani, 55, 138. +Guatemala, 205. +Guato, 142. +Guaycuru, 22. +Gudang, 24. + +Haida, 112. +Hawaii, 113, 114, 116, 117. +Head, 71. +Heap, 8, 9, 25, 70, 77, 100. +Hebrew, 86, 89, 95. +Heiltsuk, 65, 88, 163. +Herero, 150. +Hervey Islands, 118. +Hidatsa, 80, 91. +Hill End, 109. +Himalaya, 193. +Hottentot, 80, 92. +Huasteca, 204. +Hudson's Bay, 48, 61. +Hun, 169. +Hunt, Leigh, 33. + +Ibo, 185. +Icelandic, 129, 183. +Illinois, 91. +Index finger, 11, 14. +India, 96, 112, 131. +Indian, 8, 10, 13, 16, 17, 19, 32, 36, 54, 55, 59, 62, 65, 66, 79, 80, 82, + 83, 89, 90, 98, 105, 112, 171, 201. +Indian Ocean, 63, 193. +Indo-European, 76. +Irish, 129, 180. +Italian, 39, 80, 124, 129, 203. + +Jajowerong, 156. +Jallonkas, 146. +Jaloff, 146. +Japanese, 40, 86, 89, 93-95. +Java, 93, 120. +Jiviro, 61, 136. +Joints of fingers, 7, 18, 173. +Juri, 79. + +Kamassin, 130. +Kamilaroi, 27, 107, 112. +Kamtschatka, 75, 157. +Kanuri, 136, 149. +Karankawa, 68. +Karen, 112. +Keppel Bay, 24. +Ki-Nyassa, 150. +Kiriri, 55, 138, 139, 167. +Kissi, 145. +Ki-Swahili, 42. +Ki-Yau, 150. +Klamath, 58, 59. +Knot, 7, 9, 19, 40, 93, 115. +Kolyma, 75. +Kootenay, 65. +Koriak, 75. +Kredy, 149. +Kru, 146. +Ku-Mbutti, 78. +Kunama, 151. +Kuri, 188. +Kusaie, 78, 80. +Kwakiutl, 45. + +Labillardiere, 85. +Labrador, 29. +Lake Kopperamana, 107. +Latin, 40, 44, 76, 81, 86, 124, 128, 168, 169, 181, 182. +Lazi, 189. +Left hand, 10-17, 54. +Leper's Island, 16. +Leptscha, 193. +Lifu, 143. +Little finger, 10-18, 48, 54, 61, 91. +Logone, 186. +London, 124. +Lower California, 29. +Luli, 118. +Lutuami, 164. + +Maba, 80. +Macassar, 93. +Machine, Calculating, 19, 20. +Mackenzie River, 157. +Macuni, 174. +Madagascar, 8, 9. +Maipures, 15, 56. +Mairassis, 10. +Malagasy, 83, 95. +Malanta, 96. +Malay, 8, 45, 90, 93, 170. +Mallicolo, 152. +Manadu, 93. +Mandingo, 186. +Mangareva, 114. +Manx, 180. +Many, 2, 21-23, 25, 28, 100. +Maori, 64, 93, 122. +Marachowie, 26. +Mare, 84. +Maroura, 106. +Marquesas, 93, 114, 115. +Marshall Islands, 122, 152. +Massachusetts, 91, 159. +Mathematician, 2, 3, 35, 102, 127, 210. +Matibani, 151. +Matlaltzinca, 166. +Maya, 45, 46, 199, 205. +Mbayi, 111. +Mbocobi, 22. +Mbousha, 66. +Melanesia, 16, 22, 28, 84, 95. +Mende, 186. +Mexico, 29, 195, 201, 204, 208. +Miami, 91. +Micmac, 90, 160. +Middle finger, 12, 15, 62. +Mille, 122. +Minnal Yungar, 26. +Minsi, 162. +Mississaga, 44, 92. +Mississippi, 125. +Mocobi, 119. +Mohegan, 91. +Mohican, 172. +Mokko, 149. +Molele, 164. +Moneroo, 109. +Mongolian, 8. +Montagnais, 53, 54, 175. +Moree, 24. +Moreton Bay, 108. +Mort Noular, 107. +Mosquito, 69, 70, 121. +Mota, 95, 153. +Mpovi, 152. +Multiplication, 19, 33, 40, 43, 59. +Mundari, 193. +Mundo, 186. +Muralug, 17. +Murray River, 106, 109. +Muysca, 206. + +Nagranda, 207. +Nahuatl, 141, 144, 177, 201, 205. +Nakuhiva, 116, 130. +Negro, 8, 9, 15, 29, 184. +Nengone, 63, 136. +New, 128-130. +New Caledonia, 154. +New Granada, 195. +New Guinea, 10, 152. +New Hebrides, 155. +New Ireland, 45. +New Zealand, 123. +Nez Perces, 65, 158. +Ngarrimowro, 110. +Niam Niam, 64, 136. +Nicaragua, 80. +Nicobar, 78, 193. +Nightingale, 4. +Nootka, 163, 198. +Norman River, 24. +North America, 28, 82, 171, 173, 176, 194, 201. +Notch, 7, 9, 93. +Numeral frame, 19. +Nupe, 149, 186. +Nusqually, 96. + +Oceania, 115, 176. +Octonary, chap. v. +Odessa, 34. +Ojibwa, 84, 159. +Okanaken, 88. +Omaha, 161. +Omeo, 110. +Oregon, 58, 195. +Orejone, 23. +Orinoco, 54, 56, 195. +Ostrich, 71, 72. +Otomac, 15. +Otomi, 165, 199. +Ottawa, 159. +Oyster Bay, 79. + +Pacific, 29, 113, 116, 117, 131. +Palm (of the hand), 12, 14, 15. +Palm Island, 156. +Pama, 136, 155. +Pampanaga, 66. +Papaa, 148. +Paraguay, 55, 71, 118, 195. +Parana, 119. +Paris, 182. +Pawnee, 91, 121, 162. +Pebble, 7-9, 19, 40, 93, 179. +Peno, 2. +Peru (Peruvian), 2, 22, 61, 206. +Philippine, 66. +Philology (Philologist), 128, 209, 210. +Phoenician, 90, 179. +Pigmy, 69, 70, 78. +Pikumbul, 57, 138. +Pines, Isle of, 153. +Pinjarra, 26. +Plenty, 25, 77. +Point Barrow, 30, 51, 64, 83, 137, 159. +Polynesia, 22, 28, 118, 130, 170. +Pondo, 33. +Popham Bay, 107. +Port Darwin, 109. +Port Essington, 24, 107. +Port Mackay, 26. +Port Macquarie, 109. +Puget Sound, 96. +Puri, 22, 92. + +Quappa, 171, 172. +Quaternary, chap. v. +Queanbeyan, 24. +Quiche, 205. +Quichua, 61. + +Rapid, 163. +Rarotonga, 114. +Richmond River, 109. +Right hand, 10-18, 54. +Right-handedness, 13, 14. +Ring finger, 15. +Rio Grande, 195. +Rio Napo, 22. +Rio Norte, 136, 199. +Russia (Russian), 30, 35. + +Sahaptin, 158. +San Antonio, 136. +San Blas, 79, 80. +Sanskrit, 40, 92, 97, 128. +Sapibocone, 84, 167. +Saste (Shasta), 165. +Scratch, 7. +Scythian, 169. +Seed, 93. +Semitic, 89. +Senary, chap. v. +Sesake, 136, 155. +Several, 22. +Sexagesimal, 124, 208. +Shawnoe, 160. +Shell, 7, 19, 70, 93. +Shushwap, 88. +Siberia, 29, 30, 187, 190. +Sierra Leone, 83. +Sign language, 6. +Sioux, 83. +Slang, 124. +Slavonic, 40. +Snowy River, 110. +Soussou, 83, 147. +South Africa, 4, 15, 28. +South America, 2, 15, 22, 23, 27-29, 54, 57, 72, 76, 78, 79, 104, 110, 173, + 174, 194, 201, 206. +Spanish, 2, 23, 42. +Splint, 7. +Stick, 7, 179. +Stlatlumh, 88. +Streaky Bay, 26. +String, 7, 9, 64, 71. +Strong's Island, 78. +Subtraction, 19, 44-47. +Sunda, 120. +Sweden (Swedish), 129, 132, 183. + +Tacona, 2. +Taensa, 44. +Tagala, 66. +Tahiti, 114. +Tahuata, 115. +Tama, 111. +Tamanac, 54, 135. +Tambi, 120. +Tanna, 154. +Tarascan, 165. +Tariana, 174. +Tasmania, 24, 27, 79, 104, 106. +Tawgy, 130. +Tchetchnia, 188. +Tchiglit, 157, 196. +Tembu, 33. +Temne, 148. +Ternary, chap. v. +Terraba, 172. +Teutonic, 40, 41, 43, 179, 181, 208. +Texas, 69. +Thibet, 96. +Thumb, 10-18, 54, 59, 61, 62, 113, 173. +Thusch, 189. +Ticuna, 168. +Timukua, 165. +Tlingit, 136, 163, 197. +Tobi, 156. +Tonga, 33, 85. +Torres, 17, 96, 104, 105. +Totonaco, 205. +Towka, 78. +Triton's Bay, 152. +Tschukshi, 156, 191. +Tsimshian, 86, 164, 198. +Tweed River, 26. + +Uainuma, 122. +Udi, 188. +Uea, 67, 153. +United States, 29, 83, 195. +Upper Yarra, 110. +Ureparapara, 153. + +Vaturana, 96. +Vedda, 28, 31, 76, 174. +Vei, 16, 147, 185. +Victoria, 156. +Vilelo, 60. + +Waiclatpu, 164. +Wales (Welsh), 35, 46, 141, 144, 177, 180. +Wallachia, 121. +Warrego, 107, 109. +Warrior Island, 107. +Wasp, 5. +Watchandie, 29, 77. +Watji, 120. +Weedookarry, 24. +Wimmera, 107. +Winnebago, 85. +Wiraduroi, 27, 108. +Wirri-Wirri, 108. +Wokke, 112. +Worcester, Mass., Schools of, 11. + +Yahua, 168. +Yaruro, 139. +Yengen, 154. +Yit-tha, 109. +Yoruba, 33, 47, 64, 70, 185. +Yucatan, 195, 201. +Yuckaburra, 26. + +Zamuco, 55, 60, 138, 139. +Zapara, 111. +Zulu, 16, 62. +Zuni, 13, 14, 48, 49, 53, 54, 60, 83, 137. + + + + + +FOOTNOTES: + + +[1] Brinton, D.G., _Essays of an Americanist_, p. 406; and _American Race_, +p. 359. + +[2] This information I received from Dr. Brinton by letter. + +[3] Tylor, _Primitive Culture_, Vol. I. p. 240. + +[4] _Nature_, Vol. XXXIII. p. 45. + +[5] Spix and Martius, _Travels in Brazil_, Tr. from German by H.E. Lloyd, +Vol. II. p. 255. + +[6] De Flacourt, _Histoire de le grande Isle de Madagascar_, ch. xxviii. +Quoted by Peacock, _Encyc. Met._, Vol. I. p. 393. + +[7] Bellamy, Elizabeth W., _Atlantic Monthly_, March, 1893, p. 317. + +[8] _Grundriss der Sprachwissenschaft_, Bd. III. Abt. i., p. 94. + +[9] Pruner-Bey, _Bulletin de la Societe d'Anthr. de Paris_, 1861, p. 462. + +[10] "Manual Concepts," _Am. Anthropologist_, 1892, p. 292. + +[11] Tylor, _Primitive Culture_, Vol. I. p. 245. + +[12] _Op. cit._, _loc. cit._ + +[13] "Aboriginal Inhabitants of Andaman Islands," _Journ. Anth. Inst._, +1882, p. 100. + +[14] Morice, A., _Revue d'Anthropologie_, 1878, p. 634. + +[15] Macdonald, J., "Manners, Customs, etc., of South African Tribes," +_Journ. Anthr. Inst._, 1889, p. 290. About a dozen tribes are enumerated by +Mr. Macdonald: Pondos, Tembucs, Bacas, Tolas, etc. + +[16] Codrington, R.H., _Melanesians, their Anthropology and Folk-Lore_, p. +353. + +[17] _E.g._ the Zunis. See Cushing's paper quoted above. + +[18] Haddon, A.C., "Ethnography Western Tribes Torres Strait," _Journ. +Anth. Inst._, 1889, p. 305. For a similar method, see _Life in the Southern +Isles_, by W.W. Gill. + +[19] Tylor, _Primitive Culture_, Vol. I. p. 246. + +[20] Brinton, D.G., Letter of Sept. 23, 1893. + +[21] _Ibid_. The reference for the Mbocobi, _infra_, is the same. See also +Brinton's _American Race_, p. 361. + +[22] Tylor, _Primitive Culture_, Vol. I. p. 243. + +[23] _Op. cit._, _loc. cit._ + +[24] Hyades, _Bulletin de la Societe d'Anthr. de Paris_, 1887, p. 340. + +[25] Wiener, C., _Perou et Bolivie_, p. 360. + +[26] Marcoy, P., _Travels in South America_, Vol. II p. 47. According to +the same authority, most of the tribes of the Upper Amazon cannot count +above 2 or 3 except by reduplication. + +[27] _Op. cit._, Vol. II. p. 281. + +[28] _Glossaria Linguarum Brasiliensium_. Bororos, p. 15; Guachi, p. 133; +Carajas, p. 265. + +[29] Curr, E.M., _The Australian Race_, Vol. I. p. 282. The next eight +lists are, in order, from I. p. 294, III. p. 424, III. p. 114, III. p. 124, +II. p. 344, II. p. 308, I. p. 314, III. p. 314, respectively. + +[30] Bonwick, J., _The Daily Life and Origin of the Tasmanians_, p. 144. + +[31] Latham, _Comparative Philology_, p. 336. + +[32] _The Australian Race_, Vol. I. p. 205. + +[33] Mackenzie, A., "Native Australian Langs.," _Journ. Anthr. Inst._, +1874, p. 263. + +[34] Curr, _The Australian Race_, Vol. II. p. 134. The next four lists are +from II. p. 4, I. p. 322, I. p. 346, and I. p. 398, respectively. + +[35] Curr, _op. cit._, Vol. III. p. 50. + +[36] _Op. cit._, Vol. III. p. 236. + +[37] Mueller, _Sprachwissenschaft_. II. i. p. 23. + +[38] _Op. cit._, II. i. p. 31. + +[39] Bonwick, _op. cit._, p. 143. + +[40] Curr, _op. cit._, Vol. I. p. 31. + +[41] Deschamps, _L'Anthropologie_, 1891, p. 318. + +[42] Man, E.H. _Aboriginal Inhabitants of the Andaman Islands_, p. 32. + +[43] Mueller, _Sprachwissenschaft_, I. ii. p. 29. + +[44] Oldfield, A., Tr. Eth. Soc. Vol. III. p. 291. + +[45] Bancroft, H.H., _Native Races_, Vol. I. p. 564. + +[46] "Notes on Counting, etc., among the Eskimos of Point Barrow." _Am. +Anthrop._, 1890, p. 38. + +[47] _Second Voyage_, p. 556. + +[48] _Personal Narrative_, Vol. I. p. 311. + +[49] Burton, B.F., _Mem. Anthr. Soc. of London_, Vol. I. p. 314. + +[50] _Confessions_. In collected works, Edinburgh, 1890, Vol. III. p. 337. + +[51] Ellis, Robert, _On Numerals as Signs of Primeval Unity_. See also +_Peruvia Scythia_, by the same author. + +[52] Stanley, H.M., _In Darkest Africa_, Vol. II. p. 493. + +[53] Stanley, H.M., _Through the Dark Continent_, Vol. II. p. 486. + +[54] Haumonte, Parisot, Adam, _Grammaire et Vocabulaire de la Langue +Taensa_, p. 20. + +[55] Chamberlain, A.F., _Lang. of the Mississaga Indians of Skugog. Vocab._ + +[56] Boas, Fr., _Sixth Report on the Indians of the Northwest_, p. 105. + +[57] Beauregard, O., _Bulletin de la Soc. d'Anthr. de Paris_, 1886, p. 526. + +[58] Ray, S.H., _Journ. Anthr. Inst._, 1891, p. 8. + +[59] _Op. cit._, p. 12. + +[60] Mueller, _Sprachwissenschaft_, IV. i. p. 136. + +[61] Brinton, _The Maya Chronicles_, p. 50. + +[62] Trumbull, _On Numerals in Am. Ind. Lang._, p. 35. + +[63] Boas, Fr. This information was received directly from Dr. Boas. It has +never before been published. + +[64] Bancroft, H.H., _Native Races_, Vol. II. p. 753. See also p. 199, +_infra_. + +[65] Mann, A., "Notes on the Numeral Syst. of the Yoruba Nation," _Journ. +Anth. Inst._, 1886, p. 59, _et seq._ + +[66] Mueller, _Sprachwissenschaft_, IV. i. p. 202. + +[67] Trumbull, J.H., _On Numerals in Am. Ind. Langs._, p. 11. + +[68] Cushing, F.H., "Manual Concepts," _Am. Anthr._, 1892, p. 289. + +[69] Grimm, _Geschichte der deutschen Sprache_, Vol. I. p. 239. + +[70] Murdoch, J., _American Anthropologist_, 1890, p. 39. + +[71] Kleinschmidt, S., _Grammatik der Groenlandischen Sprache_, p. 37. + +[72] Brinton, _The Arawak Lang. of Guiana_, p. 4. + +[73] Petitot, E., _Dictionnaire de la langue Dene-Dindjie_, p. lv. + +[74] Gilij, F.S., _Saggio di Storia Am._, Vol. II. p. 333. + +[75] Mueller, _Sprachwissenschaft_, II. i. p. 389. + +[76] _Op. cit._, p. 395. + +[77] Mueller, _Sprachwissenschaft_, II. i. p. 438. + +[78] Peacock, "Arithmetic," in _Encyc. Metropolitana_, 1, p. 480. + +[79] Brinton, D.G., "The Betoya Dialects," _Proc. Am. Philos. Soc._, 1892, +p. 273. + +[80] Ridley, W., "Report on Australian Languages and Traditions." _Journ. +Anth. Inst._, 1873, p. 262. + +[81] Gatschet, "Gram. Klamath Lang." _U.S. Geog. and Geol. Survey_, Vol. +II. part 1, pp. 524 and 536. + +[82] Letter of Nov. 17, 1893. + +[83] Mueller, _Sprachwissenschaft_, II. i. p. 439. + +[84] Hale, "Indians of No. West. Am.," _Tr. Am. Eth. Soc._, Vol. II. p. 82. + +[85] Brinton, D.G., _Studies in So. Am. Native Languages_, p. 25. + +[86] _Tr. Am. Philological Association_, 1874, p. 41. + +[87] Tylor, _Primitive Culture_, Vol. I. p. 251. + +[88] Mueller, _Sprachwissenschaft_, IV. i. p. 27. + +[89] See _infra_, Chapter VII. + +[90] Ellis, A.B., _Ewe Speaking Peoples_, etc., p. 253. + +[91] Tylor, _Primitive Culture_, Vol. I. p. 256. + +[92] Stanley, _In Darkest Africa_, Vol. II. p. 493. + +[93] Chamberlain, A.F., _Proc. Brit. Ass. Adv. of Sci._, 1892, p. 599. + +[94] Boas, Fr., "Sixth Report on Northwestern Tribes of Canada," _Proc. +Brit. Ass. Adv. Sci._, 1890, p. 657. + +[95] Hale, H., "Indians of Northwestern Am.," _Tr. Am. Eth. Soc._, Vol. II. +p. 88. + +[96] _Op. cit._, p. 95. + +[97] Mueller, _Sprachwissenschaft_, II. ii. p. 147. + +[98] Schoolcraft, _Archives of Aboriginal Knowledge_, Vol. IV. p. 429. + +[99] Du Chaillu, P.B., _Tr. Eth. Soc._, London, Vol. I. p. 315. + +[100] Latham, R.G., _Essays, chiefly Philological and Ethnographical_, p. +247. The above are so unlike anything else in the world, that they are not +to be accepted without careful verification. + +[101] Pott, _Zaehlmethode_, p. 45. + +[102] Gatschet, A.S., _The Karankawa Indians, the Coast People of Texas_. +The meanings of 6, 7, 8, and 9 are conjectural with me. + +[103] Stanley, H.M., _In Darkest Africa_, Vol. II. p. 492. + +[104] Mueller, _Sprachwissenschaft_, II. i. p. 317. + +[105] Toy, C.H., _Trans. Am. Phil. Assn._, 1878, p. 29. + +[106] Burton, R.F., _Mem. Anthrop. Soc. of London_. 1, p. 314. In the +illustration which follows, Burton gives 6820, instead of 4820; which is +obviously a misprint. + +[107] Dobrizhoffer, _History of the Abipones_, Vol. II. p. 169. + +[108] Sayce, A.H., _Comparative Philology_, p. 254. + +[109] _Tr. Eth. Society of London _, Vol. III. p. 291. + +[110] Ray, S.H., _Journ. Anthr. Inst._, 1889, p. 501. + +[111] Stanley, _In Darkest Africa_, Vol. II. p. 492. + +[112] _Op. cit._, _loc. cit._ + +[113] Tylor, _Primitive Culture_, Vol. I. p. 249. + +[114] Mueller, _Sprachwissenschaft_, IV. i. p. 36. + +[115] Martius, _Glos. Ling. Brasil._, p. 271. + +[116] Tylor, _Primitive Culture_, Vol. I. p. 248. + +[117] Roth, H. Ling, _Aborigines of Tasmania_, p. 146. + +[118] Lull, E.P., _Tr. Am. Phil, Soc._, 1873, p. 108. + +[119] Ray, S.H. "Sketch of Api Gram.," _Journ. Anthr. Inst._, 1888, p. 300. + +[120] Kleinschmidt, S., _Grammatik der Groenlandischen Spr._, p. 39. + +[121] Mueller, _Sprachwissenschaft_, I. ii. p. 184. + +[122] _Op. cit._, I. ii. p. 18, and II. i. p. 222. + +[123] Squier, G.E., _Nicaragua_, Vol. II. p. 326. + +[124] Schoolcraft, H.R., _Archives of Aboriginal Knowledge_, Vol. II. p. +208. + +[125] Tylor, _Primitive Culture_, Vol. I. p. 264. + +[126] Goedel, "Ethnol. des Soussous," _Bull. de la Soc. d'Anthr. de Paris_, +1892, p. 185. + +[127] Ellis, W., _History of Madagascar_, Vol. I. p. 507. + +[128] Beauregard, O., _Bull. de la Soc. d'Anthr. de Paris_, 1886, p. 236. + +[129] Schoolcraft, H.R., _Archives of Aboriginal Knowledge_, Vol. II. p. +207. + +[130] Tylor, _Primitive Culture_, Vol. I. p. 249. + +[131] _Op. cit._ Vol. I. p. 250. + +[132] Peacock, _Encyc. Metropolitana_, 1, p. 478. + +[133] _Op. cit._, _loc. cit._ + +[134] Schoolcraft, H.R., _Archives of Aboriginal Knowledge_, Vol. II. p. +213. + +[135] _Op. cit._, p. 216. + +[136] _Op. cit._, p. 206. + +[137] Mariner, _Gram. Tonga Lang._, last part of book. [Not paged.] + +[138] Morice, A.G., "The Dene Langs," _Trans. Can. Inst._, March 1890, p. +186. + +[139] Boas, Fr., "Fifth Report on the Northwestern Tribes of Canada," +_Proc. Brit. Ass. Adv. of Science_, 1889, p. 881. + +[140] _Do. Sixth Rep._, 1890, pp. 684, 686, 687. + +[141] _Op. cit._, p. 658. + +[142] Bancroft, H.H., _Native Races_, Vol. II. p. 499. + +[143] _Tr. Ethnological Soc. of London_, Vol. IV. p. 92. + +[144] Any Hebrew lexicon. + +[145] Schroeder, P., _Die Phoenizische Sprache, _p. 184 _et seq._ + +[146] Mueller, _Sprachwissenschaft_, II. ii. p. 147. + +[147] _On Numerals in Am. Indian Languages._ + +[148] Ellis, A.B., _Ewe Speaking Peoples_, etc., p. 253. The meanings here +given are partly conjectural. + +[149] Pott, _Zaehlmethode_, p. 29. + +[150] Schoolcraft, _op. cit._, Vol. IV. p. 429. + +[151] Trumbull, _op. cit._ + +[152] Chamberlain, A.F., _Lang, of the Mississaga Indians_, Vocab. + +[153] Crawfurd, _Hist. Ind. Archipelago_, 1, p. 258. + +[154] Hale, H., _Eth. and Philol._, Vol. VII.; Wilkes, _Expl. Expedition_, +Phil. 1846, p. 172. + +[155] Crawfurd, _op. cit._, 1, p. 258. + +[156] _Op. cit._, _loc. cit._ + +[157] Bancroft, H.H., _Native Races_, Vol. II. p. 498. + +[158] Vignoli, T., _Myth and Science_, p. 203. + +[159] Codrington, R.H., _The Melanesian Languages_, p. 249. + +[160] _Op. cit._, _loc. cit._ + +[161] Codrington, R.H., _The Melanesian Languages_, p. 249. + +[162] Wickersham, J., "Japanese Art on Puget Sound," _Am. Antiq._, 1894, p. +79. + +[163] Codrington, R.H., _op. cit._, p. 250. + +[164] Tylor, _Primitive Culture_, Vol. I. p. 252. + +[165] Compare a similar table by Chase, _Proc. Amer. Philos. Soc._, 1865, +p. 23. + +[166] _Leibnitzii Opera_, III. p. 346. + +[167] Pruner-Bey, _Bulletin de la Soc. d'Anthr. de Paris_, 1860, p. 486. + +[168] Curr, E.M., _The Australian Race_, Vol. I. p. 32. + +[169] Haddon, A.C., "Western Tribes of the Torres Straits," _Journ. Anthr. +Inst._, 1889, p. 303. + +[170] Taplin, Rev. G., "Notes on a Table of Australian Languages," _Journ. +Anthr. Inst.,_ 1872, p. 88. The first nine scales are taken from this +source. + +[171] Latham, R.G., _Comparative Philology_, p. 352. + +[172] It will be observed that this list differs slightly from that given +in Chapter II. + +[173] Curr, E.M., _The Australian Race_, Vol. III. p. 684. + +[174] Bonwick, _Tasmania_, p. 143. + +[175] Lang, J.D., _Queensland_, p. 435. + +[176] Bonwick, _Tasmania_, p. 143. + +[177] Mueller, _Sprachwissenschaft_, II. i. p. 58. + +[178] _Op. cit._, II. i. p. 70. + +[179] _Op. cit._, II. i. p. 23. + +[180] Barlow, H., "Aboriginal Dialects of Queensland," _Journ. Anth. +Inst._, 1873, p. 171. + +[181] Curr, E.M., _The Australian Race_, Vol. II. p. 26. + +[182] _Op. cit._, Vol. II. p. 208. + +[183] _Op. cit._, Vol. II. p. 278. + +[184] _Op. cit._, Vol. II. p. 288. + +[185] _Op. cit._, Vol. I. p. 258. + +[186] _Op. cit._, Vol. I. p. 316. + +[187] _Op. cit._, Vol. III. p. 32. The next ten lists are taken from the +same volume, pp. 282, 288, 340, 376, 432, 506, 530, 558, 560, 588, +respectively. + +[188] Brinton, _The American Race_, p. 351. + +[189] Martius, _Glossaria Ling. Brazil._, p. 307. + +[190] _Op. cit._, p. 148. + +[191] Mueller, _Sprachwissenschaft_, II. i. p. 438. + +[192] Peacock, "Arithmetic," _Encyc. Metropolitana_, 1, p. 480. + +[193] Brinton, _Studies in So. Am. Native Langs._, p. 67. + +[194] _Op. cit._, _loc. cit._ + +[195] Brinton, _Studies in So. Am. Native Langs._, p. 67. The meanings of +the numerals are from Peacock, _Encyc. Metropolitana_, 1, p. 480. + +[196] Mason, _Journ. As. Soc. of Bengal_, Vol. XXVI. p. 146. + +[197] Curr, E.M., _The Australian Race_, Vol. III. p. 108. + +[198] Bancroft, H.H., _Native Races_, Vol. I. p. 274. + +[199] Clarke, Hyde, _Journ. Anthr. Inst._, 1872, p. clvii. In the article +from which this is quoted, no evidence is given to substantiate the +assertion made. It is to be received with great caution. + +[200] Hale, H., _Wilkes Exploring Expedition_, Vol. VII. p. 172. + +[201] _Op. cit._, p. 248. + +[202] Hale, _Ethnography and Philology, _p. 247. + +[203] _Loc. cit._ + +[204] Ellis, _Polynesian Researches_, Vol. IV. p. 341. + +[205] Gill, W.W., _Myths and Songs of the South Pacific_, p. 325. + +[206] Peacock, "Arithmetic," _Encyc. Metropolitana_, 1, p. 479. + +[207] Peacock, _Encyc. Metropolitana_, 1, p. 480. + +[208] _Sprachverschiedenheit_, p. 30. + +[209] Crawfurd, _History of the Indian Archipelago_, Vol. I. p. 256. + +[210] Pott, _Zaehlmethode_, p. 39. + +[211] _Op. cit._, p. 41. + +[212] Mueller, _Sprachwissenschaft_, II. i. p. 317. See also Chap. III., +_supra_. + +[213] Long, S.H., _Expedition_, Vol. II. p. lxxviii. + +[214] Martius, _Glossaria Ling. Brasil._, p. 246. + +[215] Hale, _Ethnography and Philology_, p. 434. + +[216] Mueller, _Sprachwissenschaft_, II. ii. p. 82. + +[217] The information upon which the above statements are based was +obtained from Mr. W.L. Williams, of Gisborne, N.Z. + +[218] _Primitive Culture_, Vol. I. p. 268. + +[219] Ralph, Julian, _Harper's Monthly_, Vol. 86, p. 184. + +[220] Lappenberg, J.M., _History of Eng. under the Anglo-Saxon Kings_, Vol. +I. p. 82. + +[221] The compilation of this table was suggested by a comparison found in +the _Bulletin Soc. Anth. de Paris_, 1886, p. 90. + +[222] Hale, _Ethnography and Philology_, p. 126. + +[223] Mueller, _Sprachwissenschaft_, II. ii. p. 183. + +[224] Bachofen, J.J., _Antiquarische Briefe_, Vol. I. pp. 101-115, and Vol. +II. pp. 1-90. + +[225] An extended table of this kind may be found in the last part of +Nystrom's _Mechanics_. + +[226] Schubert, H., quoting Robert Flegel, in Neumayer's _Anleitung zu +Wissenschaftlichen Beobachtung auf Reisen_, Vol. II. p. 290. + +[227] These numerals, and those in all the sets immediately following, +except those for which the authority is given, are to be found in Chapter +III. + +[228] Codrington, _The Melanesian Languages_, p. 222. + +[229] Mueller, _Sprachwissenschaft_, II. ii. p. 83. + +[230] _Op. cit._, I. ii. p. 55. The next two are the same, p. 83 and p. +210. The meaning given for the Bari _puoek_ is wholly conjectural. + +[231] Gallatin, "Semi-civilized Nations," _Tr. Am. Eth. Soc._, Vol. I. p. +114. + +[232] Mueller, _Sprachwissenschaft_, II. ii. p. 80. Erromango, the same. + +[233] Boas, Fr., _Proc. Brit. Ass'n. Adv. Science_, 1889, p. 857. + +[234] Hankel, H., _Geschichte der Mathematik_, p. 20. + +[235] Murdoch, J., "Eskimos of Point Barrow," _Am. Anthr._, 1890, p. 40. + +[236] Martius, _Glos. Ling. Brasil._, p. 360. + +[237] Du Graty, A.M., _La Republique du Paraguay_, p. 217. + +[238] Codrington, _The Melanesian Languages_, p. 221. + +[239] Mueller, _Sprachwissenschaft_, II. i. p. 363. + +[240] Spurrell, W., _Welsh Grammar_, p. 59. + +[241] Olmos, Andre de, _Grammaire Nahuatl ou Mexicaine_, p. 191. + +[242] Moncelon, _Bull. Soc. d'Anthr. de Paris_, 1885, p. 354. This is a +purely digital scale, but unfortunately M. Moncelon does not give the +meanings of any of the numerals except the last. + +[243] Ellis, _Peruvia Scythia_, p. 37. Part of these numerals are from +Martius, _Glos. Brasil._, p. 210. + +[244] Codrington, _The Melanesian Languages_, p. 236. + +[245] Schweinfurth, G., _Linguistische Ergebnisse einer Reise nach +Centralafrika_, p. 25. + +[246] Park, M., _Travels in the Interior Districts of Africa_, p. 8. + +[247] Pott, _Zaehlmethode_, p. 37. + +[248] _Op. cit._, p. 39. + +[249] Mueller, _Sprachwissenschaft_, IV. i. p. 101. The Kru scale, kindred +with the Basa, is from the same page. + +[250] Park, in Pinkerton's _Voyages and Travels_, Vol. XVI. p. 902. + +[251] Park, _Travels_, Vol. I. p. 16. + +[252] Schweinfurth, G., _Linguistische Ergebnisse einer Reise nach +Centralafrika_, p. 78. + +[253] Park, _Travels_, Vol. I. p. 58. + +[254] Goedel, "Ethnol. des Soussous," _Bull. Soc. Anth. Paris_, 1892, p. +185. + +[255] Mueller, _Sprachwissenschaft_, I. ii. p. 114. The Temne scale is from +the same page. These two languages are closely related. + +[256] _Op. cit._, I. ii. p. 155. + +[257] _Op. cit._, I. ii. p. 55. + +[258] Long, C.C., _Central Africa_, p. 330. + +[259] Mueller, _Sprachwissenschaft_, IV. i. p. 105. + +[260] Pott, _Zaehlmethode_, p. 41. + +[261] Mueller, _op. cit._, I. ii. p. 140. + +[262] Mueller, _Sprachwissenschaft_, IV. i. p. 81. + +[263] Pott, _Zaehlmethode_, p. 41. + +[264] Mueller, _op. cit._, I. ii., p. 210. + +[265] Pott, _Zaehlmethode_, p. 42. + +[266] Schweinfurth, _Linguistische Ergebnisse_, p. 59. + +[267] Mueller, _Sprachwissenschaft_, I. ii. p. 261. The "ten" is not given. + +[268] Stanley, _Through the Dark Continent_, Vol. II. p. 490. Ki-Nyassa, +the same page. + +[269] Mueller, _op. cit._, I. ii. p. 261. + +[270] Du Chaillu, _Adventures in Equatorial Africa_, p. 534. + +[271] Mueller, _Sprachwissenschaft_, III. i. p. 65. + +[272] Du Chaillu, _Adventures in Equatorial Africa_, p. 533. + +[273] Mueller, _op. cit._, III. ii. p. 77. + +[274] Balbi, A., _L'Atlas Eth._, Vol. I. p. 226. In Balbi's text 7 and 8 +are ansposed. _Taru_ for 5 is probably a misprint for _tana_. + +[275] Du Chaillu, _op. cit._, p. 533. The next scale is _op. cit._, p. 534. + +[276] Beauregard, O., _Bull. Soc. Anth. de Paris_, 1886, p. 526. + +[277] Pott, _Zaehlmethode_, p. 46. + +[278] _Op. cit._, p. 48. + +[279] Turner, _Nineteen Years in Polynesia_, p. 536. + +[280] Erskine, J.E., _Islands of the Western Pacific_, p. 341. + +[281] _Op. cit._, p. 400. + +[282] Codrington, _Melanesian Languages_, pp. 235, 236. + +[283] Peacock, _Encyc. Met._, Vol. 1. p. 385. Peacock does not specify the +dialect. + +[284] Erskine, _Islands of the Western Pacific_, p. 360. + +[285] Turner, G., _Samoa a Hundred Years Ago_, p. 373. The next three +scales are from the same page of this work. + +[286] Codrington, _Melanesian Languages_, p. 235. The next four scales are +from the same page. Perhaps the meanings of the words for 6 to 9 are more +properly "more 1," "more 2," etc. Codrington merely indicates their +significations in a general way. + +[287] Hale, _Ethnography and Philology_, p. 429. The meanings of 6 to 9 in +this and the preceding are my conjectures. + +[288] Mueller, _Sprachwissenschaft_, IV. i. p. 124. + +[289] Aymonier, E., _Dictionnaire Francaise-Cambodgien_. + +[290] Mueller, _Op. cit._, II. i. p. 139. + +[291] Mueller, _Sprachwissenschaft_, II. i. p. 123. + +[292] Wells, E.R., Jr., and John W. Kelly, Bureau of Ed., Circ. of Inf., +No. 2, 1890. + +[293] Pott, _Zaehlmethode_, p. 57. + +[294] Mueller, _Op. cit._, II. i. p. 161. + +[295] Petitot, _Vocabulaire Francaise Esquimau_, p. lv. + +[296] Mueller, _Sprachwissenschaft_, II. i. p. 253. + +[297] Mueller, _Op. cit._, II. I. p. 179, and Kleinschmidt, _Groenlandisches +Grammatik_. + +[298] Adam, L., _Congres Int. des Am._, 1877, p. 244 (see p. 162 _infra_). + +[299] Gallatin, "Synopsis of Indian Tribes," _Trans. Am. Antq. Soc._, 1836, +p. 358. The next fourteen lists are, with the exception of the Micmac, from +the same collection. The meanings are largely from Trumbull, _op. cit._ + +[300] Schoolcraft, _Archives of Aboriginal Knowledge_, Vol. II. p. 211. + +[301] Schoolcraft, _Archives of Aboriginal Knowledge_, Vol. V. p. 587. + +[302] In the Dakota dialects 10 is expressed, as here, by a word signifying +that the fingers, which have been bent down in counting, are now +straightened out. + +[303] Boas, _Fifth Report B.A.A.S._, 1889. Reprint, p. 61. + +[304] Boas, _Sixth Report B.A.A.S._, 1890. Reprint, p. 117. Dr. Boas does +not give the meanings assigned to 7 and 8, but merely states that they are +derived from 2 and 3. + +[305] _Op. cit._, p. 117. The derivations for 6 and 7 are obvious, but the +meanings are conjectural. + +[306] Boas, _Sixth Report B.A.A.S._, 1889. Reprint, pp. 158, 160. The +meanings assigned to the Tsimshian 8 and to Bilqula 6 to 8 are conjectural. + +[307] Hale, _Ethnography and Philology_, p. 619. + +[308] _Op. cit._, _loc. cit._ + +[309] Hale, _Ethnography and Philology_, p. 619. + +[310] Mueller, _Sprachwissenschaft_, II. i. p. 436. + +[311] _Op. cit._, IV. i. p. 167. + +[312] _Op. cit._, II. i. p. 282. + +[313] _Op. cit._, II. i. p. 287. The meanings given for the words for 7, 8, +9 are conjectures of my own. + +[314] Mueller, _Sprachwissenschaft_, II. i. p. 297. + +[315] Pott, _Zaehlmethode_, p. 90. + +[316] Mueller, _op. cit._, II. i. p. 379. + +[317] Gallatin, "Semi-Civilized Nations of Mexico and Central America," +_Tr. Am. Ethn. Soc._, Vol. I. p. 114. + +[318] Adam, Lucien, _Congres Internationale des Americanistes_, 1877, Vol. +II. p. 244. + +[319] Mueller, _Sprachwissenschaft_, II. i. p. 395. I can only guess at the +meanings of 6 to 9. They are obviously circumlocutions for 5-1, 5-2, etc. + +[320] _Op. cit._, p. 438. Mueller has transposed these two scales. See +Brinton's _Am. Race_, p. 358. + +[321] Marcoy, P., _Tour du Monde_, 1866, 2eme sem. p. 148. + +[322] _Op. cit._, p. 132. The meanings are my own conjectures. + +[323] An elaborate argument in support of this theory is to be found in +Hervas' celebrated work, _Arithmetica di quasi tutte le nazioni +conosciute_. + +[324] See especially the lists of Hale, Gallatin, Trumbull, and Boas, to +which references have been given above. + +[325] Thiel, B.A., "Vocab. der Indianier in Costa Rica," _Archiv fuer +Anth._, xvi. p. 620. + +[326] These three examples are from A.R. Wallace's _Narrative of Travels on +the Amazon and Rio Negro_, vocab. Similar illustrations may be found in +Martius' _Glos. Brasil_. + +[327] Martius, _Glos. Brasil._, p. 176. + +[328] Adam, L., _Congres International des Americanistes_, 1877, Vol. II. +p. 244. Given also _supra_, p. 53. + +[329] O'Donovan, _Irish Grammar_, p. 123. + +[330] Armstrong, R.A., _Gaelic Dict._, p. xxi. + +[331] Spurrell, _Welsh Dictionary_. + +[332] Kelly, _Triglot Dict._, pub. by the Manx Society. + +[333] Guillome, J., _Grammaire Francaise-Bretonne_, p. 27. + +[334] Groeber, G., _Grundriss der Romanischen Philologie_, Bd. I. p. 309. + +[335] Pott, _Zaehlmethode_, p. 88. + +[336] Van Eys, _Basque Grammar_, p. 27. + +[337] Pott, _Zaehlmethode_, p. 101. + +[338] _Op. cit._, p. 78. + +[339] Mueller, _Sprachwissenschaft_, I. ii. p. 124. + +[340] _Op. cit._, p. 155. + +[341] _Op. cit._, p. 140. + +[342] _Op. cit._, _loc. cit._ + +[343] Schweinfurth, _Reise nach Centralafrika_, p. 25. + +[344] Mueller, _Sprachwissenschaft_, IV. i. p. 83. + +[345] _Op. cit._, IV. i. p. 81. + +[346] _Op. cit._, I. ii. p. 166. + +[347] Long, C.C., _Central Africa_, p. 330. + +[348] Peacock, _Encyc. Met._, Vol. I. p. 388. + +[349] Mueller, _Sprachwissenschaft_, III. ii. p. 64. The next seven scales +are from _op. cit._, pp. 80, 137, 155, 182, 213. + +[350] Pott, _Zaehlmethode_, p. 83. + +[351] _Op. cit._, p. 83,--Akari, p. 84; Circassia, p. 85. + +[352] Mueller, _Sprachwissenschaft_, II. i. p. 140. + +[353] Pott, _Zaehlmethode_, p. 87. + +[354] Mueller, _Sprachwissenschaft_, II. ii. p. 346. + +[355] _Op. cit._, III. i. p. 130. + +[356] Man, E.H., "Brief Account of the Nicobar Islands," _Journ. Anthr. +Inst._, 1885, p. 435. + +[357] Wells, E.R., Jr., and Kelly, J.W., "Eng. Esk. and Esk. Eng. Vocab.," +Bureau of Education Circular of Information, No. 2, 1890, p. 65. + +[358] Petitot, E., _Vocabulaire Francaise Esquimau_, p. lv. + +[359] Boas, Fr., _Proc. Brit. Ass. Adv. Sci._, 1889, p. 857. + +[360] Boas, _Sixth Report on the Northwestern Tribes of Canada_, p. 117. + +[361] Boas, Fr., _Fifth Report on the Northwestern Tribes of Canada_, p. +85. + +[362] Gallatin, _Semi-Civilized Nations_, p. 114. References for the next +two are the same. + +[363] Bancroft, H.H., _Native Races of the Pacific States_, Vol. II. p. +763. The meanings are from Brinton's _Maya Chronicles_, p. 38 _et seq._ + +[364] Brinton, _Maya Chronicles_, p. 44. + +[365] Simeon Remi, _Dictionnaire de la langue nahuatl_, p. xxxii. + +[366] An error occurs on p. xxxiv of the work from which these numerals are +taken, which makes the number in question appear as 279,999,999 instead of +1,279,999,999. + +[367] Gallatin, "Semi-Civilized Nations of Mexico and Central America," +_Tr. Am. Ethn. Soc._ Vol. I. p. 114. + +[368] Pott, _Zaehlmethode_, p. 89. The Totonacos were the first race Cortez +encountered after landing in Mexico. + +[369] _Op. cit._, p. 90. The Coras are of the Mexican state of Sonora. + +[370] Gallatin, _Semi-Civilized Nations_, p. 114. + +[371] Humboldt, _Recherches_, Vol. II. p. 112. + +[372] Squier, _Nicaragua_, Vol. II. p. 326. + +[373] Gallatin, _Semi-Civilized Nations_, p. 57. + + + + + + +End of Project Gutenberg's The Number Concept, by Levi Leonard Conant + +*** END OF THIS PROJECT GUTENBERG EBOOK THE NUMBER CONCEPT *** + +***** This file should be named 16449.txt or 16449.zip ***** +This and all associated files of various formats will be found in: + https://www.gutenberg.org/1/6/4/4/16449/ + +Produced by Jonathan Ingram, Hagen von Eitzen and the +Online Distributed Proofreading Team at https://www.pgdp.net + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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