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A HISTORY OF MATHEMATICS

ANTIQUITY

NUMBER-SYSTEMS AND NUMERALS

NEARLY all number-systems, both ancient and modern, are based on the scale of 5, 10, or 20. The reason for this it is not difficult to see. When a child learns to count, he makes use of his fingers and perhaps of his toes.

In the same way the savages of prehistoric times unquestionably counted on their fingers and in some cases also on their toes. Such is indeed the practice of the African, the Eskimo, and the South Sea Islander of to-day." This recourse to the fingers has often resulted in the development of a more or less extended pantomime number-system, in which the fingers were used as in a deaf and dumb alphabet. Evidence of the prevalence of finger symbolisms is found among the ancient Egyptians, Babylonians, Greeks, and Romans, as also among the Europeans of the middle ages : even now nearly all Eastern nations use finger symbolisms. The Chinese express on the left hand

1 L. L. Conant, “ Primitive Number-Systems," in Smithsonian Report, 1892, p. 584.

“all numbers less than 100,000; the thumb nail of the right hand touches each joint of the little finger, passing first up the external side, then down the middle, and afterwards up the other side of it, in order to express the nine digits; the tens are denoted in the same way, on the second finger; the hundreds on the third; the thousands on the fourth; and tenthousands on the thumb. It would be merely necessary to proceed to the right hand in order to be able to extend this system of numeration.” 1

So common is the use of this fingersymbolism that traders are said to communicate to one another the price at which they are willing to buy or sell by touching hands, the act being concealed by their cloaks from observation of by-standers.

Had the number of fingers and toes been different in man, then the prevalent number-systems of the world would have been different also. We are safe in saying that had one more finger sprouted from each human hand, making twelve fingers in all, then the numerical scale adopted by civilized nations would not be the decimal, but the duodecimal. Two more symbols would be necessary to represent 10 and 11, respectively. As far as arithmetic is concerned, it is certainly to be regretted that a sixth finger did not appear. Except for the necessity of using two more signs or numerals and of being obliged to learn the multiplication table as far as 12 x 12, the duodecimal system is decidedly superior to the decimal. The number twelve has for its exact divisors 2, 3, 4, 6, while ten has only 2 and 5. In ordinary business affairs, the fractions 1, }, }, are used extensively, and it is very convenient to have a base which is an exact multiple of 2, 3, and 4. Among the most zealous advocates of the duodecimal scale was Charles XII.

1 GEORGE PEACOCK, article “ Arithmetic,” in Encyclopædia Metropolitana (The Encyclopædia of Pure Mathematics), p. 394. Hereafter we shall cite this very valuable article as PEACOCK.

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of Sweden, who, at the time of his death, was contemplating the change for his dominions from the decimal to the duodecimal. But it is not likely that the change will ever be brought about. So deeply rooted is the decimal system that when the storm of the French Revolution swept out of existence other old institutions, the decimal system not only remained unshaken, but was more firmly established than

The advantages of twelve as a base were not recognized until arithmetic was so far developed as to make a change impossible. “The case is the not uncommon one of high civilization bearing evident traces of the rudeness of its origin in ancient barbaric life." 2

Of the notations based on human anatomy, the quinary and vigesimal systems are frequent among the lower races, while the higher nations have usually avoided the one as too scanty and the other as too cumbrous, preferring the intermediate decimal system. Peoples have not always consistently adhered to any one scale. In the quinary system, 5, 25, 125, 625, etc., should be the values of the successive higher units, but a quinary system thus carried out was never in actual use: whenever it was extended to higher numbers it invariably ran either into the decimal or into the vigesimal system. “The home par excellence of the quinary, or rather of the quinaryvigesimal scale, is America. It is practically universal among the Eskimo tribes of the Arctic regions. It prevailed among a considerable portion of the North American Indian tribes, and was almost universal with the native races of Central and

1 Conant, op. cit., p. 589..

2 E. B. TYLOR, Primitive Culture, New York, 1889, Vol. I., p. 272. In some respects a scale having for its base a power of 2– the base 8 or 16, for instance, — is superior to the duodecimal, but it has the disadvantage of not being divisible by 3. See W. W. Johnson, “Octonary Numeration,” Bull. N. Y. Math. Soc., 1891, Vol. I., pp. 1-6.

3 TYLOR, op. cit., Vol. I., p. 262.

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