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.' 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.' 2 The Bororos of Brazil reckon: 1. couai; 2. macouai; 3. ouai; and then go on counting on their fingers, repeating this ouai.3 Of course it no more follows among savages than among ourselves that, because a man counts

1 Sicard, 'Théorie des Signes pour l'Instruction des Sourds-Muets,' vol. ii. p. 634.

2 Stanbridge in 'Tr. Eth. Soc.' vol. i. p. 304.

3 Martius, 'Gloss. Brasil,' p. 15.

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.1 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 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, 1 Kracheninnikow, 'Kamtchatka,' p. 17.

keeping the other two hidden.'1 Throughout the world, the general relation between finger-counting and wordcounting 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,2 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,

1 Gumilla, 'Historia del Orenoco,' vol. iii. ch. xlv.; Pott, Zählmethode,'

P. 16.

2 The Eastern brokers have used for ages, and still use, the method of secretly indicating numbers to one another in bargaining, 'by snipping fingers under a cloth.' 'Every joynt and every finger hath his signification,' as an old traveller says, and the system seems a more or less artificial development of ordinary finger-counting, the thumb and little finger stretched out, and the other fingers closed, standing for 6 or 60, the addition of the fourth finger making 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 farther, &c., comes out just as in chaffering in words.

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 amgnaitone, which being translated means 'a whole hand;' 6 is expressed by a term which translates the proper gesture into words, itacond 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, itacond 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. Digitnumerals 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.'1 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 arkanek-pingasut, 'on the first foot three;' 18 is arfersanek-pingasut, on the other foot three;' when they reach 20, they can say inuk návdlugo, ‘a man ended,' or inúp avatai návdlugit, 'the man's outer members ended;' in this way by counting several men they reach higher

1 Gilij; 'Saggio di Storia Americana,' vol. ii. p. 332 (Tamanac, Maypure). Martius, 'Gloss. Brasil,' (Cayriri, Tupi, Carib, Omagua, Juri, Guachi, Coretu, Cherentes, Maxuruna, Caripuna, Cauixana, Carajás, Coroado, &c.); Dobrizhoffer, 'Abipones,' vol. ii. p. 168; Humboldt, 'Monumens,' pl. xliv. (Muysca).