A History of Japanese Mathematics/Chapter 1

by David Eugene Smith, Yoshio Mikami

CHAPTER I.

The Earliest Period.

The history of Japanese mathematics, from the most remote times to the present, may be divided into six fairly distinct periods. Of these the first extended from the earliest ages to 552^[1], a period that was influenced only indirectly if at all by Chinese mathematics. The second period of approximately a thousand years (332-1600) was characterized by the influx of Chinese learning, first through Korea and then direct from China itself, by some resulting native development, and by a season of stagnation comparable to the Dark Ages of Europe. The third period was less than a century in duration, extending from about 1600 to the beginning of Seki’s influence (about 1675). This may be called the Renaissance period of Japanese mathematics, since it saw a new and vigorous importation of Chinese science, the revival of native interest through the efforts of the immediate predecessors of Seki, and some slight introduction of European learning through the early Dutch traders and through the Jesuits. The fourth period, also about a century in length (1675 to 1775) may be compared to the synchronous period in Europe. Just as the initiative of Descartes, Newton, and Leibnitz prepared the way for the labors of the Bernoullis, Euler, Laplace, D’Alembert, and their contemporaries of the eighteenth century, so the work of the great Japanese teacher, Seki, and of his pupil Takebe, made possible a noteworthy development of the wasan^[2] of Japan during the same century. The fifth period, which might indeed be joined with the fourth, but which differs from it much as the nineteenth century of European mathematics differs from the eighteenth, extended from 1775 to 1868, the date of the opening of Japan to the Western World. This is the period of the culmination of native Japanese mathematics, as influenced more or Jess by the European learning that managed to find some entrance through the Dutch trading station at Nagasaki and through the first Christian missionaries. The sixth and final period begins with the opening of Japan to intercourse with other countries and extends to the present time, a period of marvelous change in government, in ideals, in art, in industry, in education, in mathematics and the sciences generally, and in all that makes a nation great. With these stupendous changes of the present, that have led Japan to assume her place among the powers of the world, there has necessarily come both loss and gain. Just as the world regrets the apparent submerging of the exquisite native art of Japan in the rising tide of commercialism, so the student of the history of mathematics must view with sorrow the necessary decay of the wasan and the reduction or the elevation of this noble science to the general cosmopolitan level. The mathematics of the present in Japan is a broader science than that of the past; but it is no longer Japanese mathematics,—it is the mathematics of the world.

It is now proposed to speak of the first period, extending from the most remote times to 552. From the nature of the case, however, little exact information can be expected of this period, It is like seeking for the early history of England from native sources, excluding all information transmitted through Roman writers. Egypt developed a literature in very remote times, and recorded it upon her monuments and upon papyrus rolls, and Babylon wrote her records upon both stone and clay; but Japan had no early literature, and if she possessed any ancient written records they have long since perished.

It was not until the fifteenth year of the Emperor Ōjin (284), so the story goes, that Chinese ideograms, making their way through Korea, were first introduced into Japan. Japanese nobles now began to learn to read and write, a task of enormous difficulty in the Chinese system. But the records themselves have long since perished, and if they contained any knowledge of mathematics, or if any mathematics from China at that time reached the shores of Japan, all knowledge of this fact has probably gone forever. Nevertheless there is always preserved in the language of a people a great amount of historical material, and from this and from folklore and tradition we can usually derive some little knowledge of the early life and customs and number-science of any nation.

So it is with Japan. There seems to have been a number mysticism there as in all other countries. There was the usual reaching out after the unknown in the study of the stars, of the elements, and of the essence of life and the meaning of death. The general expression of wonder that comes from the study of number, of forms, and of the arrangements of words and objects, is indicated in the language and the traditions of Japan as in the language and traditions of all other peoples. Thus we know that the Jindai monji, “letters of the era of the gods”,^[3] go back to remote times, and this suggests an early cabala, very likely with its usual accompaniment of number values to the letters; but of positive evidence of this fact we have none, and we are forced to rely at present only upon conjecture.^[4]

Practically only one definite piece of information has come 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^[5] 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^[6] 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

The interesting features of the ancient system are the decimal system and the use of the word yorozu, which now means 10000. This, however, may be a meaning that came with the influx of Chinese learning, and we are not at all certain that in ancient Japanese it stood for the Greek myriad^[7]. The use of yorozu for 10000 was adopted in later times when the number names came to be based upon Chinese roots, and it may possibly have preceded the entry of Chinses learning in historic times. Thus 10^[8] was not “hundred thousand”^[9] in this later period, but “ten myriads”,^[10] and our million^[11] is a hundred myriads.^[8] Now this system of numeration by myriads is one of the frequently observed evidences of early intercourse between the scholars of the East and the West. Trades people and the populace at large did not need such large numbers, but to the scholar they were significant. When, therefore, we find the myriad as the base of the Greek system,^[12] and find it more or less in use in India,^[13] and know that it still persists in China,^[14] and see systematically used in the ancient Japanese system as well as in the modern number names, we are convinced that there must have been a considerable intercourse of scholars at an early date.^[15]

Of the rest of Japanese mathematics in this early period we are wholly ignorant, save that we know a little of the ancient system of measures and that a calendar existed. How the merchants computed, whether the almost universal finger computation of ancient peoples had found its way so far to the East, what was known in the way of mensuration, how much of a crude primitive observation of the movements of the stars was carried on, what part was played by the priest in the orientation of shrines and temples, what was the mystic significance of certain numbers, what, if anything, was done in the recording of numbers by knotted cords, or in representing them by symbols,—all these things are looked for in the study of any primitive mathematics, but they are looked for in vain in the evidences thus far at hand with respect to the earliest period of Japanese history, It is to be hoped that the spirit of investigation that is now so manifest in Japan will result in throwing more light upon this interesting period in which mathematics took its first root upon Japanese soil.

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1. All dates are expressed according to the Christian calendar and are to be taken as after Christ unless the contrary is stated.
2. The native mathematics, from Wa (Japan) and sen (mathematics). The word is modern, having been employed to distinguish the native theory from the western mathematics, the yōsan.
3. Nothing definite is known as to these letters, They may have been different alphabetic forms. Monji (or moji) means letters, Jin is god, and dai is the age or era. The expression may also be rendered “letters of the age of heros”, using the term hero to mean a mythological semi-divinity, as it is used in early Greek lore.
4. There is here, howewer, an excellent field for some Japanese scholar to search the native folklore for new material. Our present knowledge of the Jindai comes chiefly from a chapter in the Nihon-gi (Records of Japan) entitled Jindai wo Maki (Records of the Gods’ Age), written by Prince Toneri Shinnō in 720. This is probably based upon early legends handed down by the Kataribe, a class of men who in ancient times transmitted the legends orally, somewhat like the old English bards.
5. Ψαμμίτης, De karena numero, as it appears in Basel edition of 1544.
6. 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.
7. Μυρίοι, 10 000.
8. Mono yorozu, or, in modern Japanese, hyaku man.
9. Which would, if so considered, appear as momo chi, or in modern Japanese as hyaku sen.
10. So yorozu, a softened form of tō yorozu. In modern Japanese, jiu man, man being the myriad.
11. Mille + on, “big thousand”, just as saloon is salle + on, a big hall, and gallon is gill + on, a big gill.
12. See, for example, Gow, J., History of Greek Mathematics, Cambridge 1884, and similar works.
13. See Colebrooke, H. T., Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhaseara. London 1817, p. 4; Taylor, J., Lilawati. Bombay 1816, p. 5.
14. Williams, S. W., The Middle Kingdom. New York 1882; edition of 1895, vol. I, p. 619. Thus Wan sui is a myriad of years, and Wan sui Yeh means the Lord of a Myriad Years, i.e., the Emperor. The swastika is used by the Buddhists in China as a symbol for myriad. The use of the myriad in China is very ancient.
15. There is considerable literature upon this subject, and it deserves even more attention, See, for example, the following: Klingsmill, T. W., The Intercourse of China with Eastern Turkestan … in the second century B. C., in the Journal of the Royal Asiatic Society, N. S. London 1882, vol XIV, p. 74. A Japanese scholar, T. Kimura, is just at present maintaining that his people have a common ancestry with the races of the Greco-Roman civilization, basing his belief upon a comparison of the mythology and the language of the two civilizations. See also P. von Bohlen, Das alte Indien wit besonderer Ricksicht auf Ægypten. Konigsberg 1830; Reinaud, Relations politiques ef commerciales l’ Empire Romain avec l’ Asie orientale, Paris 1863; P. A. di San Filipo, Delle Relazioni antiche et moderne fra L’ italia e l’ India, Rome 1886; Smith and Karpinski, The Hindu-Arabic Numerals. Boston 1911, with extensive bibliography on this point.

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