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4

I. The Earliest Period.

down to us concerning the very early mathematics of Japan, and this relates to the number system. Tradition tells us that in the reign of Izanagi-no-Mikoto, the ancestor of the Mikados, long before the unbroken dynasty was founded by Jimmu (660 B. C.), a system of numeration was known that extended to very high powers of ten, and that embodied essentially the exponential law used by Archimedes in his Sand Reckoner^[1] that

aman=am+n.

In this system the number names were not those of the present, but the system may have been the same, although modern Japanese anthropologists have serious doubts upon this matter. The following table^[2] has been given as representing the ancient system, and it is inserted as a possibility, but the whole matter is in need of further investigation:

Ancient		Modern	Ancient		Modern
1	hito	ichi	100	momo	hyaku
2	futa	ni	1000	chi	sen
3	mi	san	10 000	yorozu	man
4	yo	shi	100 000	so yorozu	jiu man
5	itsu	go	1 000 000	momo yorozu	hyaku man
6	mu	roku	10 000 000	chi yorozu	sen man
7	nana	shichi	100 000 000	yorozu yorozu	oku
8	ya	hachi	1 000 000 000	so yorozu yorozu	jiu oku
9	koko	ku
10	tō	jiu

↑ Ψαμμίτης, De karena numero, as it appears in Basel edition of 1544.
↑ Endō, T., Dai Nihon Sūgaku Shi (History of Japanese mathematics, in Japanese. Tokio 1896, Book I, pp. 3—5; hereafter referred to as Endō). See also Knott, C. G. The Abacus in its historic and scientific aspects, in the Transactions of the Asiatic Society of Japan, Yokohama 1886, vol. XIV, p. 38; hereafter referred to as Knott. Another interesting form of counting is still in use in Japan, and is more closely connected with the ancient one that is the common form above given. It is as follows: (1) hitotsu, (2) futatsu, (3) mittsu, (4) yottsu, (5) itsutsu, (6) muttsu, (7) nanatsu, (8) yattsu, (9) kokonotsu, (10) tō. Still another form at present in use, and also related to the ancient one, is as follows: (1) hi, (2) fu, (3) mi, (4) yō, (5) itsu, (6) mū, (7) nana, (8) ya, (9) kono, (10) tō. Each of these forms is used only in counting, not in naming numbers, and their persistence may be compared with the “counting out” rhymes of Europe and America. It should be added that the modern forms given above are from Chinese roots.

[1] Ψαμμίτης, De karena numero, as it appears in Basel edition of 1544.

[2] Endō, T., Dai Nihon Sūgaku Shi (History of Japanese mathematics, in Japanese. Tokio 1896, Book I, pp. 3—5; hereafter referred to as Endō). See also Knott, C. G. The Abacus in its historic and scientific aspects, in the Transactions of the Asiatic Society of Japan, Yokohama 1886, vol. XIV, p. 38; hereafter referred to as Knott. Another interesting form of counting is still in use in Japan, and is more closely connected with the ancient one that is the common form above given. It is as follows: (1) hitotsu, (2) futatsu, (3) mittsu, (4) yottsu, (5) itsutsu, (6) muttsu, (7) nanatsu, (8) yattsu, (9) kokonotsu, (10) tō. Still another form at present in use, and also related to the ancient one, is as follows: (1) hi, (2) fu, (3) mi, (4) yō, (5) itsu, (6) mū, (7) nana, (8) ya, (9) kono, (10) tō. Each of these forms is used only in counting, not in naming numbers, and their persistence may be compared with the “counting out” rhymes of Europe and America. It should be added that the modern forms given above are from Chinese roots.

[1]

[2]