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Ideas of Number derived from experience--State of Arithmetic among un

civilized races-Small extent of Numeral-words among low tribesCounting 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,' 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.': 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 :

1 Mariner, ‘Tonga Islands,' vol. ii. p. 390.

... 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.

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."

1 Crowther, ' Yoruba Vocab.'; Burton, ‘W. & W. from W. Africa,' p. 253. O daju danu, o ko mo essan messan.-You (may seem) very clever, (but) you can't tell 9 X 9.'

Low in ' Journ. Ind. Archip.' vol. i. p. 408 ; ‘Year-Books Edw. l.' (xx.-i.) ed. Horwood, p. 220.

* Spix and Martius, · Reise in Brazilien,' p. 387.


In a Puri vocabulary the numerals are given as I. omi; 2. curiri ; 3. prica, ‘many': in a Botocudo vocabulary, I. 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. 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-taura (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: I. mal ; 2. bularr ; 3. guliba ; 4. bularr-bularr, ‘twotwo'; 5. bulaguliba, 'two-three'; 6. guliba-guliba ' threethree.' These Australian examples are at least evidence of a very scanty as well as clumsy numeral system among certain tribes. 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,

1* Tasmanian Journal,' vol. i. ; Backhouse, ‘Narr.' p. 104; Milligan in • Papers, &c., Roy. Soc. Tasmania,' vol. iii. part ii. 1859.

* Oldfield in ‘Tr. Eth. Soc.'; vol. iii. p. 291 ; Lang, 'Queensland,' p. 433 ; *Latham, Comp. Phil.'p. 352. Other terms in Bonwick, l. c.

' 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.': The Bororos of Brazil reckon : 1. couai; 2. macouai ; 3. ouai ; and then go on counting on their fingers, repeating this ouai.: 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. ü. p. 634.

* Stanbridge in ‘Tr. Eth. Soc.' vol. i. p. 304. 8 Martius, Gloss. Brasil,' p. 15.

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