Popular Science Monthly/Volume 80/June 1912/Chinese Mathematics

by David Eugene Smith

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CHINESE MATHEMATICS

By Professor DAVID EUGENE SMITH

TEACHERS COLLEGE, COLUMBIA UNIVERSITY

NO one who is interested in China and in things Chinese, and no one to whom the evolution of thought appeals, can fail to appreciate the recent articles by Dr. Edmunds, the learned president of the Canton Christian College, upon science among the Chinese.^[1] His extensive travels in all portions of the country, his own scientific attainments, his wide acquaintance with Chinese scholars, his connection with numerous scientific expeditions, and his official position at the head of one of the most progressive colleges in the country, all qualify him to speak as one having authority.

It is, however, quite natural that one whose tastes are not primarily in the line of mathematics should fail to do justice to the work of Chinese scholars in this field. It is true that this work was not of a high order, and yet it must be said that it ranked with that which was being done in other branches of science, and had not the relatively low standing that would be inferred from Dr. Edmund's statements. These statements are summarized in the following:

The study of arithmetic has attracted attention among the Chinese from early times, and notices found in historical works indicate some treatises^[2] extant even in the Han Dynasty (206 B.C.-A.D. 214), followed by a great number of general and particular works down to the Sung Dynasty (1020-1120 A.D.). The Hindu processes in algebra were known to Chinese mathematicians, but though studied even after intercourse between the countries had ceased, these branches made slow progress down to the end of the Ming Dynasty (A.D. 1368-1644).^[3]

Now as a matter of fact there is a good deal known of the mathematics of China in the pre-Christian era, and in certain respects their algebra in the Middle Ages was much in advance of that of their European contemporaries. Furthermore, this algebra appears to have been indigenous to China. While Sanskrit was known there very early, and by about 800 A.D. was even taught in Japan (through the writings of the great scholar, Kōbō Daishi), there is nothing in the mathematics of either country that shows dependence upon any known works of Hindu scholars. On the contrary, it would seem that Brahmagupta, who wrote in Ujjain in the seventh century, was indebted for at least one of his problems to the great Chinese classic, the Chiu-chang Suan-shu.

Of the scholars whose contributions to mathematics were noteworthy, it will suffice to mention only a few, although it should be stated that these are the greatest of their respective periods so far as we now know.

No one knows how far back the Chow-pi goes. It purports to be a dialogue between Chow Kung and Kao (or Shang Kao) and to have been written c. 1100 B.C. Wylie^[4] translates part of it, and shows that it contains a reference to the Pythagorean proposition and to a primitive trigonometry. It has long been known that the discovery of Pythagoras was the proof and not merely the fact, for the latter is mentioned in Egyptian writings before his time, and in Hindu works that probably antedated him, so that it is not surprising to find it in China.

Chang T'sang, who died in 153 b.c., restored the Chiu-chang Suanshu, or Arithmetical Rules in Nine Sections, for the antiquity of which, in its original form, great claims are made.^[5]

From the standpoint of mathematics the most interesting features of this work are the use of negative numbers, the trigonometry of the right triangle, and the fang-ch'êng process. The last named constituted one of the nine sections and concerned the solution of simultaneous linear equations. This would hardly be worth mentioning except for three reasons: (1) We have nothing of this kind in the algebra of Europe as early as this; (2) from this method of the Chinese came, by direct descent, the early Japanese method that led by obvious steps to the invention of determinants by Seki before the idea occurred to Leibnitz;^[6] (3) the method for the extraction of roots led Ch'in Chiu-shao, in 1247, to anticipate Horner's method, as will presently be shown.

Sun-tsǔ, whose date is unknown, but who probably lived in the third century of the Christian era, wrote a work on arithmetic in which he set forth the process known as t'ai-yen ch'iu-yi-shu, a form of indeterminate analysis that was afterwards employed with much success. It is this general form to which we give the name of Diophantine analysis, although Diophantus probably lived after Sun-tsǔ. One of his problems is as follows: "Find a number which when divided by 3 leaves a remainder of 2; when divided by 5 leaves a remainder of 3; and when divided by 7 leaves a remainder of 2." At a considerably later date such problems were common in Europe, and were evidently imported from the East.

Tsu Ch'ung-chih (428-499)^[7] certainly deserves mention if any standing is to be accorded to Metius in the history of mathematics, since he discovered the latter's value of tt some twelve centuries before it saw light in Europe. About two hundred years before him Liu Hui^[8] (in 263 A.D.) had given the value ¹⁵⁷⁄₅₀ (=3.14), and Wang Fan had suggested ¹⁴²⁄₄₅ (=3.1555. . .). But Tsu Ch'ung-chih, working from inscribed and circumscribed polygons exactly as Archimedes had done, showed that the ratio lay between 3.1415926 and 3.1415927. As limits he fixed upon ²²⁄₇, the Archimedes superior limit, and ³⁵⁵⁄₁₁₃, the value found by Metius. How Tsu came upon these limits we do not know, since his work (the Chui-shu) is lost, but it is possible, as Wei^[9] asserts, that he knew something of infinite series.

Wang Hs'iao-t'ung, who lived in the first part of the seventh century, wrote the Ch'i-ku Suan-ching, in which appeared an approximate method of solving a numerical cubic equation. At a later period this would not be significant, but when we bear in mind that this is two centuries before Al Khowārazmi (c. 825) wrote the first book bearing the title "Algebra," and some three hundred years before Alkhazin (c. 950) and Al Mohani were working on this simple cubic, it is interesting.

The golden era of native Chinese algebra was the thirteenth century, made notable by reason of the works of three men living in widely different parts of the empire. Of these, one was Ch'in Chiu-shao,^[10] who wrote the Su-shu Chiu-chang in 1247. This must always stand out in the history of mathematics as a noteworthy contribution, for here we find the detailed solution of a numerical higher equation by the method rediscovered by Horner in 1819, the only essential difference being in the numerals employed. As already stated, Ch'in merely elaborated the process for finding the square and cube roots as laid down in the Chiu-chang Suan-shu some fourteen centuries earlier, and this raises the question. How did Leonardo Fibonacci of Pisa solve the numerical equation of which he gives the root to such a high degree of approximation? He wrote his work in 1202. Did he have some hidden knowledge that had come from the Far East—some work upon which he as well as Ch'in Chiu-shao was able to build? It is one of the many questions in the history of mathematics that still remain unanswered. That the problem of the couriers, commonly attributed to the Italians, is also found in Ch'in's work, is likewise significant.

Li Yeh^[11] (1178-1265) composed two algebras, the T'sê yüan Hai-ching (1248) and the Yi-ku Yen-tuan (1259). Curiously enough, both works relate solely to the method of stating equations from the problems proposed, and not to the method of solving these equations. He also applied algebra to trigonometry, however, thus anticipating in some measure the European analytic treatment:

Chu Shih-chieh, living also in the thirteenth century, wrote his Suan-hsiao Chi-mêng in 1299, and his Szǔ-yüan Yü-chien in 1303. In these two works the native algebra of the Chinese may be said to have culminated, the methods of his immediate predecessors being here brought to a high degree of perfection. In the latter treatise the socalled Pascal triangle is found, and Chu mentions it as an ancient device that was used in solving higher equations. This was some three hundred and fifty years before Pascal (1653) wrote upon the triangle.^[12]

Kuo Shou-ching (1231-1316) introduced the study of the spherical triangle into China, although for astronomical purposes only. His work was apparently influenced by the Arabs, and so can hardly be called a native Chinese production.

No mention has been made of a work known as the Wu tsao,^[13] written in the fifth century; of the Suan-ching, one of the great treatises on Chinese arithmetic; nor of Chin Lwan who wrote the Wu-kingsuan-shu in the seventh century; nor of his probable contemporary, Chang Kew-kien, who also wrote a Suan-ching, nor of several other well-known writers, because these men contributed nothing to the science of mathematics. They were makers of text-books with a genius for exposition, but without a genius for mathematical discovery.

Enough has been stated, however, to show that the Chinese probably found out for themselves certain truths of geometry, and among these the Pythagorean theorem; that they early developed a plane trigonometry; that they did good work in approximating the value of $\pi$; that they possibly did some original work in infinite series; and that they certainly led the world at one time in algebra. It is probable that we shall soon see the publications of translations of the writings of the early mathematicians of China, or at least such a study of their works as Endō, Hayashi, Kikuchi, Fujisawa and Mikami have made of the native Japanese treatises. When this comes to pass we may possibly be able to appreciate the contributions of Chinese scholars even more highly.^[14]

1. The Popular Science Monthly, Vol. LXXTX., p. 521; Vol. LXXX., p. 22.
2. Misprinted "treaties" in the original.
3. Vol. LXXIX., p. 527. Substantially repeated in Vol. LXXX., p. 30.
4. "Chinese Researches," Shanghai, 1897, Part III., p. 163.
5. From the preface of the edition by Liu Hui (third century A.D.) we have this statement: "After the terrible Ching had burned the books, the classics and (works on) arts were dispersed and lost. Later, in the Han Dynasty, the Duke of Pei-ping, Chang T'sang, and also Ken Shou-Ch'ang (Ching Ch'ou-ch'ang) a Commissary agent, were well known because of their talents in (the domain of) number. T'sang and this other, because of the incompleteness of the ancient writings, revised (by adding to and taking from) the work. So did each claim. Upon comparing the contents their works were seen to differ from the original, but the subject matter is much alike." The origin of the work is often asserted to go back to c. 2650 B.C.
6. See T. Hayashi, "The Fukudai and Determinants in Japanese Mathematics," in the Tōkyō Sugaku-Bulurigakkwai Kizi, Vol. V. (2), p. 254.
7. Not the sixth century, as Cantor states.
8. Lew-hwuy.
9. The Chinese historian of mathematics,
10. Tsin Kiū-tschau, as Cantor has it.
11. Li Yay, as Cantor has it.
12. It seems first to have appeared in print in a work by Apianus, 1527.
13. Five sections.
14. The writer is indebted to various works upon Chinese science, and to the help of a number of scholars. It will possibly assist some reader if a few of these authorities are mentioned: A. Wylie, "Chinese Researches," Shanghai, 1897, Pt. III., p. 159; M. Courant, "Bibliographie Coréene," Paris, 1896, Vol. III., p. 2; J. Legge, "Chinese Classics," 2d ed., Vol. I., p. 4; H. Cordier, "Bibliotheca Sinica," Paris, 1905-6, Vol. II., cols. 1372, seq.; A. Vissière, "Recherches sur l'abaque chinois," in the Bulletin de Geographie, Paris, 1892; S. W. Williams, "The Middle Kingdom," edition of 1895, Vol. I., Chap. XI.; A. Wylie, "The Mongol Astronomical Instruments in Peking," in Vol. II. of the "Travaux de la 3^e. session du Congrès internat. d. Orientalistes"; A. Wylie, "Jottings of the Science of Chinese Arithmetic," in the North China Herald for 1852; M. L. Am. Sédillot, "De l'astronomie et des mathématiques chez les Chinos," in the Boncompagni Bulletino, Vol. I., p. 161; Y. Mikami, "Mathematical Papers from the Far East," Leipzig, 1910, p. 1; Y. Mikami, "A Remark on the Chinese Mathematics in Cantor's Geschichte der Mathematik," in the Archiv der Mathematik und Physik, Bd. XV. (3), S. 68, and Bd. XVIII. (3), S. 209. There are also the various histories of mathematics, including those of Montucla (2d ed., tome I., p. 451) and Cantor (Bd. I.). The writer is also indebted to Dr. W. A. P. Martin and to Mr. Mikami for personal communications relating to the subject. He is also largely indebted to his pupil. Professor T. H. Chen, of Peking, for numerous translations, including extracts from the Chinese historian of mathematics, Mei Wuh-ngan, and a translation of the entire T'sê yüan Hai-ching (1248) of Li Yeh.