A History of Japanese Mathematics/The Second Period

by David Eugene Smith, Yoshio Mikami

CHAPTER II.

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, and 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^[1] that his religion found its way into Japan.^[2] The date usually assigned to this introduction in 552, when an image of Buddha was set up in the court of the Mikado; but evidence^[3] has been found which leads to the belief that in the sixteenth year of Ketai Tenno (an emperor who reigned in Japan from 507 to 531), that is in the year 522, a certain man named Szŭ-ma Ta^[4] came from Nan-Liang^[5] 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^[6] Wang Pao-san, a master of the calendar,^[7] and Wang Pao-liang, doctor of chronology,^[8] an astrologer, crossed over from Korea and made known the Chinese chronological system. A little later a Korean priest named Kanroku^[9] crossed from his native country and presented to the Empress Suiko a set of books upon astrology and the calendar.^[10] In the twelfth year of her reign (604) almanacs were first used in Japan, and at this period Prince Shōtoku 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 Shōtoku Taishi that tradition makes him the father of Japanese arithmetic and even the inventor of the abacus.^[11] (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).^[12] While yet crown princes 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.
Fig. 1. Shōtoku Taishi, with a soroban. From a bronze statuette.

The official records show that a university system was established by the Emperor Monbu in 701, and the mathematical studies were recognized and were regulated in the higher institutions of learning. Nine Chinese works were specified, as follows:—(1) Chou-pei (Suan-ching), (2) Sun-tsu (Suan-ching), (3) Liu-chang, (4) San-k’ai Chung-ch’a, (5) Wu-t’sao (Suan-shu), (6) Hai-tao (Suan-shu), (7) Chiu-szu, (8) Chiu-chang, (9) Chui-shu.^[13] 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. Chuo-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”.^[14] 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,^[15] and is said to have had a discussion with a nobleman named Kaou, or Shang Kao,^[16] 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.^[17]

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 before 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.”^[18] This work is sometimes, but without any good reason, assigned to Sun Wu, one of the most illustrious men of the 6th century B. C.

3. Liu-Chang. This is unknown. There was a writer named Liu Hui^[19] 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 Siang-kiai-Feh-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.^[20] It is one of the standard treatises on arithmetic to 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”,^[21] 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 Swan-shu^[22] 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,^[23] who lived in 2637 B. C., caused his minister Li Show^[24] to form the Chiu-chang.^[25] 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’ou-ch’ang about a hundred years later.^[26] Both of these writers lived in the Former Han^[27] 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,^[28] and both were ministers of the emperor.

This classical work had such an effect upon the mathematics of Japan that a summary of the contents of the books or chapters of which it is composed will not be out of place. The work contained 246 problems, and these are arranged in nine sections as follows:

(1) Fang-tien, surveying. This relates to the mensuration of various plane figures, including triangles, quadrilaterals, circles, circular segments and sectors, and the annulus. It also contains some treatment of fractions.

(2) Suh-pu (Shu-poo). This treats chiefly of commercial problems solved by the “rule of three”.

(3) Shwai-f’en (Shwae-fun, Shuai-fen). This deals with partnership.

(4) Shao-kang (Shaou-kwang). This relates to the extraction of square and cube roots, the process being much like that of the present time.

(5) Shang-kung. This has reference to the mensuration of such solids as the prism, cylinder, pyramid, circular cone, frustum of a cone, tetrahedron, and wedge.

(6) Kin-shu (Kiun-shoo, Ghün-shu) treats of allegation.

(7) Ying-pu-tsu (Ying-yu, Yin-nuh). This chapter treats of “Excess and deficiency”, and follows essentially the old rule of false position.^[29]

(8) Fang-ch’êng (Fang-chêng, Fang-ching). This chapter relates to linear equations involving two or more unknown quantities, in which both positive (ching) or negative (foo) terms are employed. The following example is a type: “If 5 oxen and 2 sheep cost 10 taels of gold, and 2 oxen and 8 sheep cost 8 taels, what is the price of each?” It is probable that this chapter contains the earliest known mention of a negative quantity, and if the ancient text has not been corrupted, it places this kind of number between 2000 and 3000 B. C.

(9) Kou-ku, a term meaning a right triangle. The essential feature of this chapter is the Pythagorean theorem, which is stated as follows: “The first side and the second side being each squared and added, the square root of the sum is the hypotenuse.” One of the twenty-four problems in this section involves the equation x2+(20+14)x–2×20×1775=0, and a rule is laid down that is equivalent to the modern formula for the quadratic. If these problems were in the original text, and that text has the antiquity usually assigned to it, concerning neither of which we are at all certain, then they contain the oldest known quadratic equation. The interrelation of ancient mathematics is seen in two problems in this chapter. One is that of the reed growing 1 foot above the surface in the center of a pond 10 feet square, which just reaches the surface when drawn to the edge of the pond, it being required to find the depth of the water. The other is the problem of the broken tree that has been a stock question for four thousand years. Both of these problems are found in the early Hindu works and were among the medieval importations into Europe.

The value of π^[30] used in the “Nine Sections” is 3, as was the case generally in early times.^[31] Commentators changed this later, Liu Hui (263) giving the value 157/50, which is equivalent to 3.14.^[32]

9. Chui-shu. This is usually supposed to be Tsu Ch’ung-chih's work which has been lost and is now known only by name.

This list includes all of the important Chinese classics in mathematics that had appeared before it was made, and it shows a serious attempt to introduce the best material available into the schools of Japan at the opening of the 8th century. It seemed that the country had entered upon an era of great intellectual prosperity, but it was like the period of Charlemagne, so nearly synchronous with it—a temporary beacon in a dark night. Instead of leading scholars to the study of pure mathematics, this introduction of Chinese science, at a time when the people were not fully capable of appreciating it, seemed rather to foster a study of astrology, and mathematics degenerated into mere puzzle solving, the telling of fortunes, and the casting of horoscopes. Japan itself was given up to wars and rumors of wars. The “Nine Sections” was forgotten, and a man who actually knew arithmetic was looked upon as a genius. The samurai or noble class disdained all commercial pursuits, and ability to operate with numbers was looked upon as evidence of low birth. Professor Nitobe has given us a picture of this feudal society in his charming little book entitled Bushido, The Soul of Japan.^[33] “Children,” he says, “were brought up with utter disregard of economy. It was considered bad taste to speak of it, and ignorance of the value of different coins was a token of good breeding. Knowledge of numbers was indispensable in the mastering of forces as well as in the distribution of benefices and fiefs, but the counting of money was left to meaner hands.” Only in the Buddhist temples in Japan, as in the Christian church schools in Europe, was the lamp of learning kept burning.^[34] In each case, however, mathematics was not a subject that appealed to the religious body. A crude theology, a purposeless logic, a feeble literature,—these had some standing; but mathematicians save for calendar purposes was ever an outcast in the temple and the church, save as it occasionally found some eccentric individual to befriend it. In the period of the Ashikaga shoguns it is asserted that there hardly could be found in all Japan a man who was versed in the art of division.^[35] To divide, the merchant resorted to the process known as Shokei-zan, a scheme of multiplication^[36] which seems in some way to have served for the inverse process as well.^[37] Nevertheless the assertion that the art of division was lost during this era of constant wars is not exact. Manuscripts on the calendar, corresponding to the European compotus rolls, and belonging to the period in question, contain examples of division, and it is probable that here, as in the West, the religious communities always had someone who knew the rudiments of clendar-reckoning. (Fig. 2.)

Three names stand out during these Dark Ages as worthy of mention. The first is that of Tenjin, or Michizane, counsellor and teacher in the court of the Emperor Uda (888—898).

Japanese Calendar Rolls.

Fig. 2. Japanese Calendar Rolls.

Uda's successor, Daigo, banished him from the court and he died in 903. He was a learned man, and after his death he was canonized under the name Tenjin (Heavenly man) and was looked upon as the patron of science and letters. (See Fig. 3.) The second is that of Michinori, Lord of the province of Hyūga. His name is connected with a mathematical theory called the Keishi-zan.^[38] It seems to have been related to permutations and to have been thought of enough consequence to attract the attention of Yoshida^[39] and of his great successor Seki^[40] in the 17th century Michinori's work was written in the Hogen period (1156—1159).
Fig. 3. Tenjin, from an old bronze.

The third name is that of Genshō, a Buddhist priest in the time of Shogun Yoriye, at the opening of the 13th century. Tradition^[41] says that he was distinguished for his arithmetical powers, but so far as we know he wrote nothing and had no permanent influence upon mathematics.

Thus passes and closes a period of a thousand years, with not a single book of any merit, and without advancing the science of mathematics a single pace. Europe was backward enough, but Japan was worse. China was doing a little, India was doing more, but the Arab was accomplishing still more through his restlessness of spirit if not through his mathematical genius. The world's rebirth was approaching, and this Renaissance came to Japan at about the time that it came to Europe, accompanied in both cases by a grafting of foreign learning upon native stock.

---

1. The Shinshiu or “True Sect” of Buddhists place his death as early as 949 B. C., but the Singalese Buddhists place it at 543 B. C., Rhys Davids, who has done so much to make Buddhism known to English readers, gives 412 B. C., and Max Müller makes it 477 B. C., See also Sumner, J., Buddhism and traditions concerning its introduction into Japan, Transactions of the Asiatic Society of Japan, Yokohama 1886, vol. XIV, p. 73. He gives the death of Buddha as 544 B. C.
2. It was introduced into China in 64 A. D., and into Korea in 372.
3. See Sumner, loc. cit., p. 78.
4. In Japanese, Shiba Tatsu.
5. I. e., South Liang, Liang being one of the southern monarchies.
6. I. e., in 554, or possibly 553.
7. In Europe he would have had charge of the Compotus, the science of the Church calender, in a Western monastery.
8. Also called a Doctor of Yih. The doctrine of Yih (changes) is set forth in the Yih King (Book of Changes), one of the ancient Five Classics of the Chinese. There is a very extensive literature upon this subject.
9. Or Ch’ūan-lo.
10. Sumner, loc. cit., p. 80, gives the date as 593. Endō, who is the leading Japanese authority, gives it as 602.
11. That this is without foundation will appear in Chapter III. The soroban which he holds in the illustration here given is an anachronism.
12. Murray, D., The Story of Japan. N. Y. 1894, p. 398, from the official records.
13. Endō, Book I, pp. 12—13.
14. Pi means leg, thigh, thigh-bone.
15. Chi Tan, known as Chow Kung (that is, the Duke of Chow), was brother and advisor to the Emperor Wu Wang of the Chow dynasty. It is possible that he wrote to Chow Li, “Institutions of the Chow Dynasty”, although it is more probable it was written for him. The establishment and prosperity of the Chow dynasty is largely due to him. There is no little as to the antiquity of this work, and the critical study of scholars may eventually place it much later than the traditional date here given.
16. Also written Shang Kaou.
17. For a translation of the dialogue see Wylie, A., Chinese Researches. Shanghai 1897, Part III p. 163.
18. His result is 23. For a method of solving see Wylie, loc. cit., p. 175.
19. Lew•hwuy according to Wylie's transliteration, who also assigns him to about the 5th century B. C.
20. But see Wylie, loc. cit., who refers it to about the 5th century, and improperly states that Wu-t’sao is the author's name. He gives it the common name of Swan-king (Arithmetical classic).
21. Also written Hae-taou-swan-king.
22. Kew chang-swan-shu, Kiu-chang-san-suh, Kieou chang.
23. Or Hoan-ti, the “Yellow Emperor”. Some writers give the date much earlier.
24. Or Li-shou.
25. Wylie, A., Jottings of the Science of Chinese Arithmetic, North China Herald for 1852, Shanghai Almanac for 1853, Chinese Researches, Shanghai 1897, Part III, page 159; Biernatzki, Die Arithmetik der Chinesen, Crelle's Journal for 1856, vol. 52.
26. For this information we are indebted to the testimony of Liu Hui, whose commentary was written in 263.
27. Also known as the Western Han.
28. Legge, J., The Chinese Classics. Oxford 1893, 2nd edition, vol. I, p. 4.
29. The Regula falsi or Regula positionis of the Middle Ages in Europe. The rule seems to have been of oriental origin.
30. In Chinese Chou-le; in Japanese yenshū ritsu.
31. It is also found in the Chou-pei, No. 1 in this list.
32. Mikami, Y., On Chinese Circle-Squarers, in the Bibliotheca Mathematica, 1910, vol. X (3), p. 193.
33. Tokio 1905, p. 88. Some historical view of these early times is given in an excellent work by W. H. Sharp, The Educational System of Japan. Bombay 1906, pp. 1, 10, 11.
34. Notably in the case of the labors of the learned Kōbō Daishi, founder of the Chēnyen sect of Buddhists, who was born in 774 A. D. See Professor T. Tanimoto's address of Kōbō Daishi. Kobe 1907.
35. Endō, Book I, p. 30.
36. Uchida Gokan, Kokon Sankwan, 1832, preface.
37. This is the opinion of Murai Chūzen who loved in the 18th century. See his Sampō Dōshi-mon. 1781. Book I, article on the origin of arithmetic.
38. Endō, Book I, p. 28; Murai Chuzen, Sampō Dōshimon.
39. See his Jinkō-ki of 1627.
40. See Chapter IV.
41. See Isomura Kitioku, Shushuo Ketsugishō, 1684, Book 4, marginal note. Isomura died in 1710.

---