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THE POPULAR SCIENCE MONTHLY

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[1] (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.[2]

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,[3] 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 ${\displaystyle \pi }$; 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

1. Li Yay, as Cantor has it.
2. It seems first to have appeared in print in a work by Apianus, 1527.
3. Five sections.