Primitive culture, vol. 1 (of 2)

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 the probable 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:—

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 beyond two. The Watchandie scale of notation is co-ote-on (one), u-tau-ra (two), bool-tha (many), and bool-tha-bat (very many). If absolutely required to express the numbers three or four, they say u-tar-ra coo-te-oo to indicate the former number, and u-tar-ra u-tar-ra to 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 that they 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 this ouai.[307] Of course it no more follows among savages than among ourselves that, because a man counts on 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 the hand which they held up to denote it, that in like manner they said two hands or half a man to denote 10, that the word foot carried on the reckoning up to 15, and to 20, which they described in words as in gesture by the hands and feet together, or as one 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 word amgnaitòne, 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 as amgna aceponàre ‘both hands.’ To denote 11 they stretch out both the hands, and adding the foot they say puittaponà tevinitpe ‘one to the foot,’ and thus up to 15, which is iptaitòne ‘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 connected with words for ‘hand,’ &c., as when the Omagua uses pua, ‘hand,’ for 5, and reduplicates this into upapua for

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 Juri ghomen apa ‘one man,’ stands for 5; the Cayriri ibichó is 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, counted quihicha 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 is tatdlimat, which there is some ground for supposing to have once meant ‘hand;’ 6 is arfinek-attausek, ‘on the other hand one,’ or more shortly arfinigdlit, ‘those which have on the other hand;’ 7 is arfinek-mardluk, ‘on the other hand two;’ 13 is arkanck-pingasut, ‘on the first foot three;’ 18 is arfersanek-pingasut, ‘on the other foot three;’ when they reach 20, they can say inuk nâvdlugo, ‘a man ended,’ or inûp avatai nâvdlugit,’ the man’s outer members ended;’ in this way by counting several men they reach higher numbers, thus expressing, for example, 53 as inûp 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 by macuilli; and as ma (ma-itl) means ‘hand,’ and cuiloa ‘to paint or depict,’ it is likely that the word for 5 may have meant something like ‘hand-depicting.’ In 10, matlactli, the word ma, ‘hand,’ appears again, while tlactli means 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.

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 with puggana, ‘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 as ome re rue tubenine, apparently ‘the two sides’ (i.e. both hands), 20 as sa re ngome,’one man,’ &c.; thus in John v. 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 is lima or rima, ‘hand,’ and the connexion is not lost by the phonetic variations among different branches of this family of languages, as in Malagasy dimy, Marquesan fima, Tongan nima, but while lima and 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 forms lima-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 an actual 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 call edesanta ‘finish hand;’ then he goes on to the thumb of the right hand, and so the word tatisitupa ‘taking the thumb’ becomes a numeral for

7. 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 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. 8, and Kijangalolunje ‘keep back one finger,’ i.e. 9, lead on to kumi, 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 their own 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:—