CHAPTERVII.THE ART OF COUNTING.

Ideas of Number derived from experience—State of Arithmetic among uncivilized races—Small extent of Numeral-words among low tribes—Counting by fingers and toes—Hand-numerals show derivation of Verbal reckoning from Gesture-counting—Etymology of Numerals—Quinary, Decimal, and Vigesimal notations of the world derived from counting on fingers and toes—Adoption of foreign Numeral-words—Evidence of development of Arithmetic from a low original level of Culture.

Mr.J. S.Mill, in his ‘System of Logic,’ takes occasion to examine the foundations of the art of arithmetic. Against Dr. Whewell, who had 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, Mr. Mill asserts that ‘two and one are equal to three’ expresses merely ‘a truth known to us by early and constant experience: an inductive truth; and such truths are the foundation of the science of Number. The fundamental truths of that science all rest on the evidence of sense; they are proved by showing to our eyes and our fingers that any given number of objects, ten balls for example, may by separation and re-arrangement exhibit to our senses all the different sets of numbers the sum of which is equal to ten. All the improved methods of teaching arithmetic to children proceed on a knowledge of this fact. All who wish to carry the child’s mind along with them in learning arithmetic; all who wish to teach numbers, and not mere ciphers—now teach it through the evidence of the senses,in the manner we have described.’ Mr. Mill’s argument is taken from the mental conditions of people among whom there exists a highly advanced arithmetic. The subject is also one to be advantageously studied from the ethnographer’s point of view. The examination of the methods of numeration in use among the lower races not only fully bears out Mr. Mill’s view, that our knowledge of the relations of numbers is based on actual experiment, but it enables us to trace the art of counting to its source, and to ascertain by what steps it arose in the world among particular races, and probably among all mankind.

In our advanced system of numeration, no limit is known either to largeness or smallness. The philosopher cannot conceive the formation of any quantity so large or of any atom so small but the arithmetician can keep pace with him, and can define it in a simple combination of written signs. But as we go downwards in the scale of culture, we find that even where the current language has terms for hundreds and thousands, there is less and less power of forming a distinct notion of large numbers, the reckoner is sooner driven to his fingers, and there increases among the most intelligent that numerical indefiniteness that we notice among children—if there were not a thousand people in the street there were certainly a hundred, at any rate there were twenty. Strength in arithmetic does not, it is true, vary regularly with the level of general culture. Some savage or barbaric peoples are exceptionally skilled in numeration. The Tonga Islanders really have native numerals up to 100,000. Not content even with this, the French explorer Labillardière pressed them farther and obtained numerals up to 1000 billions, which were duly printed, but proved on later examination to be partly nonsense-words and partly indelicate expressions,[299]so that the supposed series of high numerals forms at once a little vocabulary of Tongan indecency, and a warning as to theprobable results of taking down unchecked answers from question-worried savages. In West Africa, a lively and continual habit of bargaining has developed a great power of arithmetic, and little children already do feats of computation with their heaps of cowries. Among the Yorubas of Abeokuta, to say ‘you don’t know nine times nine’ is actually an insulting way of saying ‘you are a dunce.’[300]This is an extraordinary proverb, when we compare it with the standard which our corresponding European sayings set for the limits of stupidity: the German says, ‘he can scarce count five’; the Spaniard, ‘I will tell you how many make five’ (cuantos son cinco); and we have the same saw in England:—

‘... as sure as I’m alive,And knows how many beans make five.’

‘... as sure as I’m alive,

And knows how many beans make five.’

A Siamese law-court will not take the evidence of a witness who cannot count or reckon figures up to ten; a rule which reminds us of the ancient custom of Shrewsbury, where a person was deemed of age when he knew how to count up to twelve pence.[301]

Among the lowest living men, the savages of the South American forests and the deserts of Australia, 5 is actually found to be a number which the languages of some tribes do not know by a special word. Not only have travellers failed to get from them names for numbers above 2, 3, or 4, but the opinion that these are the real limits of their numeral series is strengthened by the use of their highest known number as an indefinite term for a great many. Spix and Martius say of the low tribes of Brazil, ‘They count commonly by their finger joints, so up to three only. Any larger number they express by the word “many.”’[302]In a Puri vocabulary the numerals are given as 1.omi; 2.curiri; 3.prica, ‘many’: in a Botocudo vocabulary, 1.mokenam; 2.uruhú, ‘many.’ The numeration of the Tasmanians is, according to Jorgensen, 1.parmery; 2.calabawa; more than 2,cardia; as Backhouse puts it, they count ‘one, two, plenty;’ but an observer who had specially good opportunities, Dr. Milligan, gives their numerals up to 5,puggana, which we shall recur to.[303]Mr. Oldfield (writing especially of Western tribes) says, ‘The New Hollanders have no names for numbers beyondtwo. The Watchandie scale of notation isco-ote-on(one),u-tau-ra(two),bool-tha(many), andbool-tha-bat(very many). If absolutely required to express the numbers three or four, they sayu-tar-ra coo-te-ooto indicate the former number, andu-tar-ra u-tar-rato denote the latter.’ That is to say, their names for one, two, three, and four, are equivalent to ‘one,’ ‘two,’ ‘two-one,’ ‘two-two.’ Dr. Lang’s numerals from Queensland are just the same in principle, though the words are different: 1.ganar; 2.burla; 3.burla-ganar, ‘two-one’; 4.burla-burla, ‘two-two’;korumba, ‘more than four, much, great.’ The Kamilaroi dialect, though with the same 2 as the last, improves upon it by having an independent 3, and with the aid of this it reckons as far as 6: 1.mal; 2.bularr; 3.guliba; 4.bularr-bularr, ‘two-two’; 5.bulaguliba, ‘two-three’; 6.guliba-guliba‘three-three.’ These Australian examples are at least evidence of a very scanty as well as clumsy numeral system among certain tribes.[304]Yet here again higher forms will have to be noticed, which in one district at least carry the native numerals up to 15 or 20.

It is not to be supposed, because a savage tribe has no current words for numbers above 3 or 5 or so, that therefore they cannot count beyond this. It appears thatthey can and do count considerably farther, but it is by falling back on a lower and ruder method of expression than speech—the gesture-language. The place in intellectual development held by the art of counting on one’s fingers, is well marked in the description which Massieu, the Abbé Sicard’s deaf-and-dumb pupil, gives of his notion of numbers in his comparatively untaught childhood: ‘I knew the numbers before my instruction, my fingers had taught me them. I did not know the ciphers; I counted on my fingers, and when the number passed 10 I made notches on a bit of wood.’[305]It is thus that all savage tribes have been taught arithmetic by their fingers. Mr. Oldfield, after giving the account just quoted of the capability of the Watchandie language to reach 4 by numerals, goes on to describe the means by which the tribe contrive to deal with a harder problem in numeration.

‘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 enquiry, 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.’ Of the aborigines of Victoria, Mr. Stanbridge says: ‘They have no name for numerals above two, but by repetition they count to five; they also record the days of the moon by means of the fingers, the bones and joints of the arms and the head.’[306]The Bororos of Brazil reckon: 1.couai; 2.macouai; 3.ouai; and then go on counting on their fingers, repeating thisouai.[307]Of course it no more follows among savages than among ourselves that, because a man countson his fingers, his language must be wanting in words to express the number he wishes to reckon. For example it was noticed that when natives of Kamchatka were set to count, they would reckon all their fingers, and then all their toes, so getting up to 20, and then would ask, ‘What are we to do next?’ Yet it was found on examination that numbers up to 100 existed in their language.[308]Travellers notice the use of finger-counting among tribes who can, if they choose, speak the number, and who either silently count it upon their fingers, or very usually accompany the word with the action; nor indeed are either of these modes at all unfamiliar in modern Europe. Let Father Gumilla, one of the early Jesuit missionaries in South America, describe for us the relation of gesture to speech in counting, and at the same time bring to our minds very remarkable examples (to be paralleled elsewhere) of the action of consensus, whereby conventional rules become fixed among societies of men, even in so simple an art as that of counting on one’s fingers. ‘Nobody among ourselves,’ he remarks, ‘except incidentally, would say for instance “one,” “two,” &c., and give the number on his fingers as well, by touching them with the other hand. Exactly the contrary happens among Indians. They say, for instance, “give me one pair of scissors,” and forthwith they raise one finger; “give me two,” and at once they raise two, and so on. They would never say “five” without showing a hand, never “ten” without holding out both, never “twenty” without adding up the fingers, placed opposite to the toes. Moreover, the mode of showing the numbers with the fingers differs in each nation. To avoid prolixity, I give as an example the number “three.” The Otomacs to say “three” unite the thumb, forefinger, and middle finger, keeping the others down. The Tamanacs show the little finger, the ring finger, and the middle finger, and close the other two. The Maipures, lastly, raise the fore, middle, and ring fingers,keeping the other two hidden.’[309]Throughout the world, the general relation between finger-counting and word-counting may be stated as follows. For readiness and for ease and apprehension of numbers, a palpable arithmetic, such as is worked on finger-joints or fingers,[310]or heaps of pebbles or beans, or the more artificial contrivances of the rosary or the abacus, has so great an advantage over reckoning in words as almost necessarily to precede it. Thus not only do we find finger-counting among savages and uneducated men, carrying on a part of their mental operations where language is only partly able to follow it, but it also retains a place and an undoubted use among the most cultured nations, as a preparation for and means of acquiring higher arithmetical methods.

Now there exists valid evidence to prove that a child learning to count upon its fingers does in a way reproduce a process of the mental history of the human race; that in fact men counted upon their fingers before they found words for the numbers they thus expressed; that in this department of culture, Word-language not only followed Gesture-language, but actually grew out of it. The evidence in question is principally that of language itself, which shows that, among many and distant tribes, men wanting to express 5 in words called it simply by their name for thehandwhich they held up to denote it, that in like manner they saidtwo handsorhalf a manto denote 10, that the wordfootcarried on the reckoning up to 15,and to 20, which they described in words as in gesture by thehands and feettogether, or asone man, and that lastly, by various expressions referring directly to the gestures of counting on the fingers and toes, they gave names to these and intermediate numerals. As a definite term is wanted to describe significant numerals of this class, it may be convenient to call them ‘hand-numerals’ or ‘digit-numerals.’ A selection of typical instances will serve to make it probable that this ingenious device was not, at any rate generally, copied from one tribe by another or inherited from a common source, but that its working out with original character and curiously varying detail displays the recurrence of a similar but independent process of mental development among various races of man.

Father Gilij, describing the arithmetic of the Tamanacs on the Orinoco, gives their numerals up to 4: when they come to 5, they express it by the wordamgnaitòne, which being translated means ‘a whole hand;’ 6 is expressed by a term which translates the proper gesture into words,itaconò amgnaponà tevinitpe‘one of the other hand,’ and so on up to 9. Coming to 10, they give it in words asamgna aceponàre‘both hands.’ To denote 11 they stretch out both the hands, and adding the foot they saypuittaponà tevinitpe‘one to the foot,’ and thus up to 15, which isiptaitòne‘a whole foot.’ Next follows 16, ‘one to the other foot,’ and so on to 20,tevin itòto, ‘one Indian;’ 21,itaconò itòto jamgnàr bonà tevinitpe‘one to the hands of the other Indian;’ 40,acciachè itòto, ‘two Indians;’ thence on to 60, 80, 100, ‘three, four, five Indians,’ and beyond if needful. South America is remarkably rich in such evidence of an early condition of finger-counting recorded in spoken language. Among its many other languages which have recognizable digit-numerals, the Cayriri, Tupi, Abipone, and Carib rival the Tamanac in their systematic way of working out ‘hand,’ ‘hands,’ ‘foot,’ ‘feet,’ &c. Others show slighter traces of the same process, where, for instance, the numerals 5 or 10 are found to be connectedwith words for ‘hand,’ &c., as when the Omagua usespua, ‘hand,’ for 5, and reduplicates this intoupapuafor 10. In some South American languages a man is reckoned by fingers and toes up to 20, while in contrast to this, there are two languages which display a miserably low mental state, the man counting only one hand, thus stopping short at 5; the Jurighomen apa‘one man,’ stands for 5; the Cayririibichóis used to mean both ‘person’ and 5. Digit-numerals are not confined to tribes standing, like these, low or high within the limits of savagery. The Muyscas of Bogota were among the more civilized native races of America, ranking with the Peruvians in their culture, yet the same method of formation which appears in the language of the rude Tamanacs is to be traced in that of the Muyscas, who, when they came to 11, 12, 13, countedquihicha ata,bosa,mica,i.e., ‘foot one, two, three.’[311]To turn to North America, Cranz, the Moravian missionary, thus describes about a century ago the numeration of the Greenlanders. ‘Their numerals,’ he says, ‘go not far, and with them the proverb holds that they can scarce count five, for they reckon by the five fingers and then get the help of the toes on their feet, and so with labour bring out twenty,’ The modern Greenland grammar gives the numerals much as Cranz does, but more fully. The word for 5 istatdlimat, which there is some ground for supposing to have once meant ‘hand;’ 6 isarfinek-attausek, ‘on the other hand one,’ or more shortlyarfinigdlit, ‘those which have on the other hand;’ 7 isarfinek-mardluk, ‘on the other hand two;’ 13 isarkanck-pingasut, ‘on the first foot three;’ 18 isarfersanek-pingasut, ‘on the other foot three;’ when they reach 20, they can sayinuk nâvdlugo, ‘a man ended,’ orinûp avatai nâvdlugit,’ the man’s outer members ended;’ in this way by counting several men they reach highernumbers, thus expressing, for example, 53 asinûp pinga-jugsâne arkanek-pingasut, ‘on the third man on the first foot three.’[312]If we pass from the rude Greenlanders to the comparatively civilized Aztecs, we shall find on the Northern as on the Southern continent traces of early finger-numeration surviving among higher races. The Mexican names for the first four numerals are as obscure in etymology as our own. But when we come to 5 we find this expressed bymacuilli; and asma(ma-itl) means ‘hand,’ andcuiloa‘to paint or depict,’ it is likely that the word for 5 may have meant something like ‘hand-depicting.’ In 10,matlactli, the wordma, ‘hand,’ appears again, whiletlactlimeans half, and is represented in the Mexico picture-writings by the figure of half a man from the waist upward; thus it appears that the Aztec 10 means the ‘hand-half’ of a man, just as among the Towka Indians of South America 10 is expressed as ‘half a man,’ a whole man being 20. When the Aztecs reach 20 they call itcempoalli, ‘one counting,’ with evidently the same meaning as elsewhere, one whole man, fingers and toes.

Among races of the lower culture elsewhere, similar facts are to be observed. The Tasmanian language again shows the man stopping short at the reckoning of himself when he has held up one hand and counted its fingers; this appears by Milligan’s list before mentioned, which ends withpuggana, ‘man,’ standing for 5. Some of the West Australian tribes have done much better than this, using their word for ‘hand,’marh-ra;marh-jin-bang-ga, ‘half the hands,’ is 5;marh-jin-bang-ga-gudjir-gyn, ‘half the hands and one,’ is 6, and so on;marh-jin-belli-belli-gudjir-jina-bang-ga, ‘the hand on either side and half the feet,’ is 15.[313]As an example from the Melanesian languages the Maré will serve; it reckons 10 asome re rue tubenine, apparently ‘the twosides’ (i.e. both hands), 20 assa re ngome,’one man,’ &c.; thus in Johnv.5 ‘which had an infirmity thirty and eight years,’ the numeral 38 is expressed by the phrase, ‘one man and both sides five and three.’[314]In the Malayo-Polynesian languages, the typical word for 5 islimaorrima, ‘hand,’ and the connexion is not lost by the phonetic variations among different branches of this family of languages, as in Malagasydimy, Marquesanfima, Tongannima, but whilelimaand its varieties mean 5 in almost all Malayo-Polynesian dialects, its meaning of ‘hand’ is confined to a much narrower district, showing that the word became more permanent by passing into the condition of a traditional numeral. In languages of the Malayo-Polynesian family, it is usually found that 6, &c., are carried on with words whose etymology is no longer obvious, but the formslima-sa,lima-zua‘hand-one,’ ‘hand-two,’ have been found doing duty for 6 and 7.[315]In West Africa, Kölle’s account of the Vei language gives a case in point. These negroes are so dependent on their fingers that some can hardly count without, and their toes are convenient as the calculator squats on the ground. The Vei people and many other African tribes, when counting, first count the fingers of their left hand, beginning, be it remembered, from the little one, then in the same manner those of the right hand, and afterwards the toes. The Vei numeral for 20,mō bánde, means obviously ‘a person (mo) is finished (bande),’ and similarly 40, 60, 80, &c. ‘two men, three men, four men, &c., are finished,’ It is an interesting point that the negroes who used these phrases had lost their original descriptive sense—the words have become mere numerals to them.[316]Lastly, for bringing before our minds a picture of a man counting upon his fingers, and being struck by the idea that if he describes his gestures in words, these words may become anactual name for the number, perhaps no language in the world surpasses the Zulu. The Zulu counting on his fingers begins in general with the little finger of his left hand. When he comes to 5, this he may calledesanta‘finish hand;’ then he goes on to the thumb of the right hand, and so the wordtatisitupa‘taking the thumb’ becomes a numeral for 6. Then the verbkomba‘to point,’ indicating the forefinger, or ‘pointer,’ makes the next numeral, 7. Thus, answering the question ‘How much did your master give you?’ a Zulu would say‘Ukombile’‘He pointed with his forefinger,’i.e., ‘He gave me seven,’ and this curious way of using the numeral verb is shown in such an example as‘amahasiakombile’‘the horses have pointed,’i.e., ‘there were seven of them.’ In like manner,Kijangalobili‘keep back two fingers,’i.e.8, andKijangalolunje‘keep back one finger,’i.e.9, lead on tokumi, 10; at the completion of each ten the two hands with open fingers are clapped together.[317]

The theory that man’s primitive mode of counting was palpable reckoning on his hands, and the proof that many numerals in present use are actually derived from such a state of things, is a great step towards discovering the origin of numerals in general. Can we go farther, and state broadly the mental process by which savage men, having no numerals as yet in their language, came to invent them? What was the origin of numerals not named with reference to hands and feet, and especially of the numerals below five, to which such a derivation is hardly appropriate? The subject is a peculiarly difficult one. Yet as to principle it is not altogether obscure, for some evidence is forthcoming as to the actual formation of new numeral words, these being made by simply pressing into the service names of objects or actions in some way appropriate to the purpose.

People possessing full sets of inherited numerals in theirown languages have nevertheless sometimes found it convenient to invent new ones. Thus the scholars of India, ages ago, selected a set of words from 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 its twelve signs, and ‘nail’ for 20, a word incidentally bringing in a finger notation. As Sanskrit is very rich in synonyms, and as even the numerals themselves might be used, it becomes very easy to draw up phrases or nonsense-verses to record series of numbers by this system of artificial memory. The following is a Hindu astronomical formula, a list of numbers referring to the stars of the lunar constellations. Each word stands as the mnemonic equivalent of the number placed over it in the English translation. The general principle on which the words are chosen to denote the numbers is evident without further explanation:—

‘Vahni tri rtvishu gunendu kritâgnibhûtaBânâsvinetra çara bhûku yugabdhi râmâhRudrâbdhirâmagunavedaçatâ dviyugmaDantâ budhairabhihitâh kramaço bhatârâh.’

‘Vahni tri rtvishu gunendu kritâgnibhûta

Bânâsvinetra çara bhûku yugabdhi râmâh

Rudrâbdhirâmagunavedaçatâ dviyugma

Dantâ budhairabhihitâh kramaço bhatârâh.’

3 3 6 5 3 1‘Fire, three, season, arrow, quality, moon,4 3 5four-side of die, fire, element,5 2 2 5 1 1 4 4 3Arrow, Asvin, eye, arrow, earth, earth, age, ocean, Rama,11 4 3 3 4 100 2 2Rudra, ocean, Rama, quality, Veda, hundred, two, couple,32Teeth: by the wise have been set forth in order the mighty lords.’[318]

3 3 6 5 3 1‘Fire, three, season, arrow, quality, moon,

3 3 6 5 3 1

‘Fire, three, season, arrow, quality, moon,

4 3 5four-side of die, fire, element,

4 3 5

four-side of die, fire, element,

5 2 2 5 1 1 4 4 3Arrow, Asvin, eye, arrow, earth, earth, age, ocean, Rama,

5 2 2 5 1 1 4 4 3

Arrow, Asvin, eye, arrow, earth, earth, age, ocean, Rama,

11 4 3 3 4 100 2 2Rudra, ocean, Rama, quality, Veda, hundred, two, couple,

11 4 3 3 4 100 2 2

Rudra, ocean, Rama, quality, Veda, hundred, two, couple,

32Teeth: by the wise have been set forth in order the mighty lords.’[318]

Teeth: by the wise have been set forth in order the mighty lords.’[318]

It occurred to Wilhelm von Humboldt, in studying this curious system of numeration, that he had before his eyes the evidence of a process very like that which actually produced the regular numeral words denotingone,two,three, &c., in the various languages of the world. The following passage in which, more than sixty years ago, he set forth this view, seems to me to contain a nearly perfect key to the theory of numeral words. ‘If we take into consideration the origin of actual numerals, the process of their formation appears evidently to have been the same as that here described. The latter is nothing else than a wider extension of the former. For when 5 is expressed, as in several languages of the Malay family, by “hand” (lima), this is precisely the same thing as when in the description of numbers by words, 2 is denoted by “wing.” Indisputably there lie at the root of all numerals such metaphors as these, though they cannot always be now traced. But people seem early to have felt that the multiplicity of such signs for the same number was superfluous, too clumsy, and leading to misunderstandings.’ Therefore, he goes on to argue, synonyms of numerals are very rare. And to nations with a deep sense of language, the feeling must soon have been present, though perhaps without rising to distinct consciousness, that recollections of the original etymology and descriptive meaning of numerals had best be allowed to disappear, so as to leave the numerals themselves to become mere conventional terms.

The most instructive evidence I have found bearing on the formation of numerals, other than digit-numerals, among the lower races, appears in the use on both sides of the globe of what may be called numeral-names for children. In Australia a well-marked case occurs. With all the poverty of the aboriginal languages in numerals, 3 being commonly used as meaning ‘several or many,’ the natives in the Adelaide district have for a particular purpose gone far beyond this narrow limit, and possess what is to all intents a special numeral system, extending perhaps to 9. They give fixed names to their children in order of age, which are set down as follows by Mr. Eyre: 1. Kertameru; 2. Warritya; 3. Kudnutya; 4. Monaitya; 5. Milaitya; 6. Marrutya; 7. Wangutya; 8. Ngarlaitya; 9. Pouarna. These are the male names, from which the female differ in termination. They are given at birth, more distinctive appellations being soon afterwards chosen.[319]A similar habit makes its appearance among the Malays, who in some districts are reported to use a series of seven names in order of age, beginning with 1.Sulung(‘eldest’); 2.Awang(‘friend, companion’), and ending withKechil(‘little one’), orBongsu(‘youngest’). These are for sons; daughters haveMehprefixed, and nicknames have to be used for practical distinction.[320]In Madagascar, the Malay connexion manifests itself in the appearance of a similar set of appellations given to children in lieu of proper names, which are, however, often substituted in after years. Males;Lahimatoa(‘first male’),Lah-ivo(‘intermediate male’);Ra-fara-lahy(‘last born male’). Females;Ramatoa(‘eldest female’),Ra-ivo(‘intermediate’),Ra-fara-vavy(‘last born female’).[321]The system exists inNorth America. There have been found in use among the Dacotas the following two series of names for sons and daughters in order of birth. Eldest son,Chaské; second,Haparm; third,Ha-pe-dah; fourth,Chatun; fifthHarka. Eldest daughter,Wenonah; second,Harpen; third,Harpstenah; fourth,Waska; fifth,We-harka.These mere numeral appellations they retain through childhood, till their relations or friends find occasion to replace them by bestowing some more distinctive personal name.[322]Africa affords further examples.[323]

As to numerals in the ordinary sense, Polynesia shows remarkable cases of new formation. Besides the well-known system of numeral words prevalent in Polynesia, exceptional terms have from time to time grown up. Thus the habit of altering words which sounded too nearly like a king’s name, has led the Tahitians on the accession of new chiefs to make several new words for numbers. Thus, wanting a new term for 2 instead of the ordinaryrua, they for obvious reasons took up the wordpiti, ‘together,’ and made it a numeral, while to get a new word for 5 instead ofrima, ‘hand,’ which had to be discontinued, they substitutedpae, ‘part, division,’ meaning probably division of the two hands. Such words as these, introduced in Polynesia for ceremonial reasons, are expected to be dropped again and the old ones replaced, when the reason for their temporary exclusion ceases, yet the new 2 and 5,pitiandpae, became so positively the proper numerals of the language, that they stand instead ofruaandrimain the Tahitian translation of the Gospel ofSt.John made at the time. Again, various special habits of counting in the South Sea Islands have had their effect on language. The Marquesans, counting fish or fruit by one in each hand,have come to use a system of counting by pairs instead of by units. They start withtauna, ‘a pair,’ which thus becomes a numeral equivalent to 2; then they count onward by pairs, so that when they talk oftakauor 10, they really mean 10 pair or 20. For bread-fruit, as they are accustomed to tie them up in knots of four, they begin with the wordpona, ‘knot,’ which thus becomes a real numeral for 4, and here again they go on counting by knots, so that when they saytakauor 10, they mean 10 knots or 40. The philological mystification thus caused in Polynesian vocabularies is extraordinary; in Tahitian, &c.,rauandmano, properly meaning 100 and 1,000, have come to signify 200 and 2,000, while in Hawaii a second doubling in their sense makes them equivalent to 400 and 4,000. Moreover, it seems possible to trace the transfer of suitable names of objects still farther in Polynesia in the Tongan and Maori wordtekau, 10, which seems to have been a word for ‘parcel’ or ‘bunch,’ used in counting yams and fish, as also intefuhi, 100, derived fromfuhi, ‘sheaf or bundle.’[324]

In Africa, also, special numeral formations are to be noticed. In the Yoruba language, 40 is calledogodzi, ‘a string,’ because cowries are strung by forties, and 200 isigba, ‘a heap,’ meaning again a heap of cowries. Among the Dahomans in like manner, 40 cowries make akadeor ‘string,’ 50 strings make oneafoor ‘head;’ these words becoming numerals for 40 and 2,000. When the king of Dahome attacked Abeokuta, it is on record that he was repulsed with the heavy loss of ‘two heads, twenty strings, and twenty cowries’ of men, that is to say, 4,820.[325]

Among cultured nations, whose languages are most tightly bound to the conventional and unintelligiblenumerals of their ancestors, it is likewise usual to find other terms existing which are practically numerals already, and might drop at once into the recognized places of such, if by any chance a gap were made for them in the traditional series. Had we room, for instance, for a new word instead oftwo, then eitherpair(Latinpar, ‘equal’) orcouple(Latincopula, ‘bond or tie,’) is ready to fill its place. Instead oftwenty, the good English wordscore, ‘notch,’ will serve our turn, while, for the same purpose, German can usestiege, possibly with the original sense of ‘a stall full of cattle, a sty;’ Old Norsedrôtt, ‘a company,’ Danish,snees. A list of such words used, but not grammatically classed as numerals in European languages, shows great variety: examples are, Old Norse,flockr(flock), 5;sveit, 6;drôtt(party), 20;thiodh(people), 30;fölk(people), 40;öld(people), 80;her(army), 100; Sleswig,schilk, 12 (as though we were to make a numeral out of ‘shilling’); Middle High-German,rotte, 4; New High-German,mandel, 15;schock(sheaf), 60. The Letts give a curious parallel to Polynesian cases just cited. They throw crabs and little fish three at a time in counting them, and therefore the wordmettens, ‘a throw,’ has come to mean 3; while flounders being fastened in lots of thirty, the wordkahlis, ‘a cord,’ becomes a term to express this number.[326]

In two other ways, the production of numerals from merely descriptive words may be observed both among lower and higher races. The Gallas have no numerical fractional terms, but they make an equivalent set of terms from the division of the cakes of salt which they use as money. Thustchabnana, ‘a broken piece’ (fromtchaba, ‘to break,’ as we say ‘a fraction’), receives the meaning of one-half; a term which we may compare with Latindimidium, Frenchdemi. Ordinal numbers are generally derived from cardinal numbers, asthird,fourth,fifth, fromthree,four,five. But among the very low ones there is to be seen evidence of independent formation quite unconnected with a conventional system of numerals already existing. Thus the Greenlander did not use his ‘one’ to make ‘first,’ but calls itsujugdlek, ‘foremost,’ nor ‘two’ to make ‘second,’ which he callsaipâ, ‘his companion;’ it is only at ‘third’ that he takes to his cardinals, and formspingajuatin connexion withpingasut, 3. So, in Indo-European languages, the ordinalprathamas,πρῶτος,primus,first, has nothing to do with a numerical ‘one,’ but with the prepositionpra, ‘before,’ as meaning simply ‘foremost;’ and although Greeks and Germans call the next ordinalδεύτερος,zweite, fromδυό,zwei, we call itsecond, Latinsecundus, ‘the following’ (sequi), which is again a descriptive sense-word.

If we allow ourselves to mix for a moment what is with what might be, we can see how unlimited is the field of possible growth of numerals by mere adoption of the names of familiar things. Following the example of the Sleswigers we might makeshillinga numeral for 12, and go on to express 4 bygroat;weekwould provide us with a name for 7, andcloverfor 3. But this simple method of description is not the only available one for the purpose of making numerals. The moment any series of names is arranged in regular order in our minds, it becomes a counting-machine. I have read of a little girl who was set to count cards, and she counted them accordingly, January, February, March, April. She might, of course, have reckoned them as Monday, Tuesday, Wednesday. It is interesting to find a case coming under the same class in the language of grown people. We know that the numerical value of the Hebrew letters is given with reference to their place in the alphabet, which was arranged for reasons that can hardly have had anything to do with arithmetic. The Greek alphabet is modified from a Semitic one, but instead of letting the numeral value of their letters follow throughout their newly-arranged alphabet, they reckonα, β, γ, δ, ε, properly, as 1, 2, 3, 4, 5,then put inσfor 6, and so manage to letιstand for 10, asיdoes in Hebrew, where it is really the10thletter. Now, having this conventional arrangement of letters made, it is evident that a Greek who had to give up the regular 1, 2, 3,—εἷς, δύο, τρεῖς, could supply their places at once by adopting the names of the letters which had been settled to stand for them, thus calling 1alpha, 2bēta, 3gamma, and so onward. The thing has actually happened; a remarkable slang dialect of Albania, which is Greek in structure, though full of borrowed and mystified words and metaphors and epithets understood only by the initiated, has, as its equivalent for ‘four’ and ‘ten,’ the wordsδέλταandἰῶτα.[327]

While insisting on the value of such evidence as this in making out the general principles of the formation of numerals, I have not found it profitable to undertake the task of etymologizing the actual numerals of the languages of the world, outside the safe limits of the systems of digit-numerals among the lower races, already discussed. There may be in the languages of the lower races other relics of the etymology of numerals, giving the clue to the ideas according to which they were selected for an arithmetical purpose, but such relics seem scanty and indistinct.[328]There may even exist vestiges of a growth of numerals from descriptive words in our Indo-European languages, in Hebrew and Arabic, in Chinese. Such etymologies have beenbrought forward,[329]and they are consistent with what is known of the principles on which numerals or quasi-numerals are really formed. But so far as I have been able to examine the evidence, the cases all seem so philologically doubtful, that I cannot bring them forward in aid of the theory before us, and, indeed, think that if they succeed in establishing themselves, it will be by the theory supporting them, rather than by their supporting the theory. This state of things, indeed, fits perfectly with the view here adopted, that when a word has once been taken up to serve as a numeral, and is thenceforth wanted as a mere symbol, it becomes the interest of language to allow it to break down into an apparent nonsense-word, from which all traces of original etymology have disappeared.

Etymological research into the derivation of numeral words thus hardly goes with safety beyond showing in the languages of the lower culture frequent instances of digit-numerals, words taken from direct description of the gestures of counting on fingers and toes. Beyond this, another strong argument is available, which indeed covers almost the whole range of the problem. The numerical systems of the world, by the actual schemes of their arrangement, extend and confirm the opinion that counting on fingers and toes was man’s original method of reckoning, taken up and represented in language. To count the fingers on one hand up to 5, and then go on with a secondfive, is a notation by fives, or as it is called, a quinary notation. To count by the use of both hands to 10, and thence to reckon by tens, is a decimal notation. To go on by hands and feet to 20, and thence to reckon by twenties, is a vigesimal notation. Now though in the larger proportion of known languages, no distinct mention of fingers and toes, hands and feet, is observable in the numerals themselves, yet the very schemes of quinary, decimal, and vigesimal notation remain to vouch for such hand-and-foot-counting having been the original method on which they were founded. There seems no doubt that the number of the fingers led to the adoption of the not especially suitable number 10 as a period in reckoning, so that decimal arithmetic is based on human anatomy. This is so obvious, that it is curious to see Ovid in his well-known lines putting the two facts close together, without seeing that the second was the consequence of the first.

‘Annus erat, decimum cum luna receperat orbem.Hic numerus magno tune in honore fuit.Seu quia tot digiti, per quos numerare solemus:Seu quia bis quino femina mense parit:Seu quod adusque decem numero crescente venitur,Principium spatiis sumitur inde novis.’[330]

‘Annus erat, decimum cum luna receperat orbem.

Hic numerus magno tune in honore fuit.

Seu quia tot digiti, per quos numerare solemus:

Seu quia bis quino femina mense parit:

Seu quod adusque decem numero crescente venitur,

Principium spatiis sumitur inde novis.’[330]

In surveying the languages of the world at large, it is found that among tribes or nations far enough advanced in arithmetic to count up to five in words, there prevails, with scarcely an exception, a method founded on hand-counting, quinary, decimal, vigesimal, or combined of these. For perfect examples of the quinary method, we may take a Polynesian series which runs 1, 2, 3, 4, 5, 5·1, 5·2, &c.; or a Melanesian series which may be rendered as 1, 2, 3, 4, 5,2nd1,2nd2, &c. Quinary leading into decimal is well shown in the Fellata series 1 ... 5, 5·1 ... 10, 10·1 ...10·5, 10·5·1 ... 20, ... 30, ... 40, &c. Pure decimal may be instanced from Hebrew 1, 2 ... 10, 10·1 ... 20, 20·1 ... &c. Pure vigesimal is not usual, for the obviousreason that a set of independent numerals to 20 would be inconvenient; but it takes on from quinary, as in Aztec, which may be analyzed as 1, 2 ... 5, 5·1 ... 10, 10·1 ... 10·5, 10·5·1 ... 20, 20·1 ... 20·10, 20·10·1 ... 40, &c.; or from decimal, as in Basque, 1 ... 10, 10·1 ... 20, 20·1 ... 20·10, 20·10·1 ... 40 &c.[331]It seems unnecessary to bring forward here the mass of linguistic details required for any general demonstration of these principles of numeration among the races of the world.Prof.Pott, of Halle, has treated the subject on elaborate philological evidence, in a special monograph,[332]which is incidentally the most extensive collection of details relating to numerals, indispensable to students occupied with such enquiries. For the present purpose the following rough generalization may suffice, that the quinary system is frequent among the lower races, among whom also the vigesimal system is considerably developed, but the tendency of the higher nations has been to avoid the one as too scanty, and the other as too cumbrous, and to use the intermediate decimal system. These differences in the usage of various tribes and nations do not interfere with, but rather confirm, the general principle which is their common cause, that man originally learnt to reckon from his fingers and toes, and in various ways stereotyped in language the result of this primitive method.

Some curious points as to the relation of these systems may be noticed in Europe. It was observed of a certain deaf-and-dumb boy, Oliver Caswell, that he learnt to count as high as 50 on his fingers, but always ‘fived,’ reckoning, for instance, 18 objects as ‘both hands, one hand, three fingers.’[333]The suggestion has been made that the Greek useofπεμπάζειν, ‘to five,’ as an expression for counting, is a trace of rude old quinary numeration (compare Finnishlokket‘to count,’ fromlokke‘ten’). Certainly, the Roman numeralsI, II, ... V, VI ... X, XI ... XV, XVI, &c., form a remarkably well-defined written quinary system. Remains of vigesimal counting are still more instructive. Counting by twenties is a strongly marked Keltic characteristic. The cumbrous vigesimal notation could hardly be brought more strongly into view in any savage race than in such examples as Gaelicaon deug is da fhichead‘one, ten, and two twenties,’ i.e., 51; or Welshunarbymtheg ar ugain‘one and fifteen over twenty,’ i.e., 36; or Bretonunnek ha triugent‘eleven and three twenties,’ i.e., 71. Now French, being a Romance language, has a regular system of Latin tens up to 100;cinquante,soixante,septante,huitante,nonante, which are to be found still in use in districts within the limits of the French language, as in Belgium. Nevertheless, the clumsy system of reckoning by twenties has broken out through the decimal system in France. Theseptanteis to a great extent suppressed,soixante-quatorze, for instance, standing for 74;quatre-vingtshas fairly established itself for 80, and its use continues into the nineties,quatre-vingt-treizefor 93; in numbers above 100 we findsix-vingts,sept-vingts,huit-vingts, for 120, 140, 160, and a certain hospital has its name ofLes Quinze-vingtsfrom its 300 inmates. It is, perhaps, the most reasonable explanation of this curious phenomenon, to suppose the earlier Keltic system of France to have held its ground, modelling the later French into its own ruder shape. In England, the Anglo-Saxon numeration is decimal,hund-seofontig, 70;hund-eahtatig, 80;hund-nigontig, 90;hund-teontig, 100;hund-enlufontig, 110;hund-twelftig, 120. It may be here also by Keltic survival that the vigesimal reckoning by the ‘score,’threescore and ten,fourscore and thirteen, &c., gained a position in English which it has not yet totally lost.[334]

From some minor details in numeration, ethnological hints may be gained. Among rude tribes with scanty series of numerals, combination to make out new numbers is very soon resorted to. Among Australian tribes addition makes ‘two-one,’ ‘two-two,’ express 3 and 4; in Guachi ‘two-two’ is 4; in San Antonio ‘four and two-one’ is 7. The plan of making numerals by subtraction is known in North America, and is well shown in the Aino language of Yesso, where the words for 8 and 9 obviously mean ‘two from ten,’ ‘one from ten.’ Multiplication appears, as in San Antonio, ‘two-and-one-two,’ and in a Tupi dialect ‘two-three,’ to express 6. Division seems not known for such purposes among the lower races, and quite exceptional among the higher. Facts of this class show variety in the inventive devices of mankind, and independence in their formation of language. They are consistent at the same time with the general principles of hand-counting. The traces of what might be called binary, ternary, quaternary, senary reckoning, which turn on 2, 3, 4, 6, are mere varieties, leading up to, or lapsing into, quinary and decimal methods.

The contrast is a striking one between the educated European, with his easy use of his boundless numeral series, and the Tasmanian, who reckons 3, or anything beyond 2, as ‘many,’ and makes shift by his whole hand to reach the limit of ‘man,’ that is to say, 5. This contrast is due to arrest of development in the savage, whose mind remains in the childish state which the beginning of one of our nursery number-rhymes illustrates curiously. It runs—

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