A History of Japanese Mathematics

A History of Japanese Mathematics

by David E. Smith, Yoshio Mikami

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One of the first books to show Westerners the nature of Japanese mathematics, this survey highlights the leading features in the development of the wasan, the Japanese system of mathematics. Topics include the use of the soroban, or abacus; the application of sangi, or counting rods, to algebra; the discoveries of the 17th-century sage Seki Kowa; the yenri,


One of the first books to show Westerners the nature of Japanese mathematics, this survey highlights the leading features in the development of the wasan, the Japanese system of mathematics. Topics include the use of the soroban, or abacus; the application of sangi, or counting rods, to algebra; the discoveries of the 17th-century sage Seki Kowa; the yenri, or circle principle; the work of 18th-century geometer Ajima Chokuyen; and Wada Nei's contributions to the understanding of hypotrochoids. Unabridged republication of the classic 1914 edition. 74 figures. Index.

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Dover Publications
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Dover Books on Mathematics Series
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5.40(w) x 8.50(h) x 0.60(d)

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A History of Japanese Mathematics

By David E. Smith, Yoshio Mikami

Dover Publications, Inc.

Copyright © 2004 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-15446-6


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, a period that was influenced only indirectly if at all by Chinese mathematics. The second period of approximately a thousand years (552—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 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 less 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 Ojin (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", 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.

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 that

am an = 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 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:


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. 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 Chinese learning in historic times. Thus 10 was not "hundred thousand" in this later period, but "ten myriads", and our million is a hundred myriads. 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, and find it more or less in use in India, and know that it still persists in China, and see it 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.

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.


The Second Period.

The second period in the history of Japanese mathematics (552—1600) corresponds both in time and in nature with the Dark Ages of Europe. Just as the Northern European lands came in contact with the South, and imbibed some slight draught of classical learning, and then lapsed into a state of indifference except for the influence of an occasional great soul like that of Charlemagne or of certain noble minds in the Church, so Japan, subject to the same Zeitgeist, drank lightly at the Chinese fountain and then lapsed again into semi-barbarism. Europe had her Gerbert, and Leonardo of Pisa, and Sacrobosco, but they seem like isolated beacons in the darkness of the Middle Ages; and in the same way Japan, as we shall see, had a few scholars who tended the lamp of learning in the medieval night, and who are known for their fidelity rather than for their genius.

Just as in the West we take the fall of Rome (476) and the fall of Constantinople (1453), two momentous events, as convenient limits for the Dark Ages, so in Japan we may take the introduction of Buddhism (552) and the revival of learning (about 1600) as similar limits, at least in our study of the mathematics of the country.

It was in round numbers a thousand years after the death of Buddha that his religion found its way into Japan. The date usually assigned to this introduction is 552, when an image of Buddha was set up in the court of the Mikado; but evidence has been found which leads to the belief that in the sixteenth year of Keitai Tenno (an emperor who reigned in Japan from 507 to 531), that is in the year 522, a certain man named Szu-ma Ta came from Nan-Liang in China, and set up a shrine in the province of Yamato, and in it placed an image of Buddha, and began to expound his religion. Be this as it may, Buddhism secured a foothold in Japan not far from the traditional date of 552, and two years later Wang Pao-san, a master of the calendar, and Wang Pao-liang, doctor ot chronology, an astrologer, crossed over from Korea and made known the Chinese chronological system. A little later a Korean priest named Kanroku crossed from his native country and presented to the Empress Suiko a set of books upon astrology and the calendar. In the twelfth year of her reign (604) almanacs were first used in Japan, and at this period Prince Shotoku Taishi proved himself such a fosterer of Buddhism and of learning that his memory is still held in high esteem. Indeed, so great was the fame of Sh?toku Taishi that tradition makes him the father of Japanese arithmetic and even the inventor of the abacus. (Fig. 1.)

A little later the Chinese system of measures was adopted, and in general the influence of China seems at once to have become very marked. Fortunately, just about this time, the Emperor Tenchi (Tenji) began his short but noteworthy reign (668—671). While yet crown prince this liberal-minded man invented a water clock, and divided the day into a hundred hours, and upon ascending the throne he showed his further interest by founding a school to which two doctors of arithmetic and twenty students of the subject were appointed. An observatory was also established, and from this time mathematics had recognized standing in Japan.

The official records show that a university system was established by the Emperor Monbu in 701, and that mathematical studies were recognized and were regulated in the higher institutions of learning. Nine Chinese works were specified, as follows:—(1) Chou-pei (Suall-ching), (2) Sun-tsu (Suan-ching), (3) Liu-chung, (4) San-k'ai Chung-ch'a, (5) Wu-t'sao (Suan-shu), (6) Hai-tao (Suanz-shu), (7) Chiu-szu, (8) Chiu-chang, (9) Chui-shu. Of these works, apparently the most famous of their time, the third, fourth, and seventh are lost. The others are probably known, and although they are not of native Japanese production they so greatly influenced the mathematics of Japan as to deserve some description at this time. We shall therefore consider them in the order above given.

1. Chou-pei Suan-ching. This is one of the oldest of the Chinese works on mathematics, and is commonly known in China as Chow-pi, said to mean the "Thigh bone of Chow". The thigh bone possibly signifies, from its shape, the base and altitude of a triangle. Chow is thought to be the name of a certain scholar who died in 1105 B. C., but it may have been simply the name of the dynasty. This scholar is sometimes spoken of as Chow Kung, and is said to have had a discussion with a nobleman named Kaou, or Shang Kao, which is set forth in this book in the form of a dialogue. The topic is our so-called Pythagorean theorem, and the time is over five hundred years before Pythagoras gave what was probably the first scientific proof of the proposition. The work relates to geometric measures and to astronomy.

2. Sun-tsu Suan-ching. This treatise consists of three books, and is commonly known in China as the Swan-king (Arithmetical classic) of Sun-tsu (Sun-tsze, or Swen-tse), a writer who lived probably in the 3d century A. D., but possibly much earlier. The work attracted much attention and is referred to by most of the later writers, and several commentaries have appeared upon it. Sun-tsu treats of algebraic quantities, and gives an example in indeterminate equations. This problem is to "find a number which, when divided by 3 leaves a remainder of 2, when divided by 5 leaves 3, and when divided by 7 leaves 2." This work is sometimes, but without any good reason, assigned to Sun Wu, one of the most illustratious men of the 6th century B. C.

3. Liu-Chang. This is unknown. There was a writer named Liu Hui who wrote a treatise entitled Chung-ch'a, but this seems to be No. 4 in the list.

4. San-k'ai Chung-ch'a. This is also unknown, but is perhaps Liu Hui's Chung-ch'a-keal-tsih-wang-chi-shuh (The whole system of measuring by the observation of several beacons), published in 263. The author also wrote a commentary on the Chiu-chang (No. 8 in this list). It relates to the mensuration of heights and distances, and gives only the rules without any explanation. About 1250 Yang Hway published a work entitled Siang-kiai-Kew-chang-Swan-fa (Explanation of the arithmetic of the Nine Sections), but this is too late for our purposes. He also wrote a work with a similar title Siangkiai-Jeh-yung-Swan-fa (Explanation of arithmetic for daily use).

5. Wu-t'sao Suan-shu. The author and the date of this work are both unknown, but it seems to have been written in the 2d or 3d century. It is one of the standard treatises on arithmetic of the Chinese.

6. Hai-tao Suan-shu. This was a republication of No. 4, and appeared about the time of the Japanese decree of 701. The name signifies "The Island Arithmetical Classic", and seems to come from the first problem, which relates to the measuring of an island from a distant point.

7. Chiu-szu. This work, which was probably a commentary on the Suan-shu (Swan-king) of No. 8, is lost.

8. Chiu-chang. Chiu-chang Suan-shu means "Arithmetical Rules in Nine Sections". It is the greatest arithmetical classic of China, and tradition assigns to it remote antiquity. It is related in the ancient Tung-kien-kang-muh (General History of China) that the Emperor Hwang-ti, who lived in 2637 B. C., caused his minister Li Show to form the Chiu-chang. Of the text of the original work we are not certain, for the reason that during the Ch'in dynasty (220—205 B. C.) the emperor Chi Hoang-ti decreed, in 213 B. C., that all the books in the empire should be burned. And while it is probable that the classics were all surreptitiously preserved, and while they could all have been repeated from memory, still the text may have been more or less corrupted during the reign of this oriental vandal. The text as it comes to us is that of Chang T'sang of the second century B. C., revised by Ching Ch'ouch' ang about a hundred years later. Both of these writers lived in the Former Han dynasty (202 B. C.—24 A. D.), a period corresponding in time and in fact with the Augustan age in Europe, and one in which great effort was made to restore the lost classics, and both were ministers of the emperor.


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