6oo TEE 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 un- 

 answered. That the problem of the couriers, commonly attributed to 

 the Italians, is also found in Ch'in's work, is likewise significant. 



Li Yeh" (1178-1265) composed two algebras, the T'se yiian Hai- 

 ching (1248) and the Yi-ku Yen-tuan (1259). Curiously enough, both 

 works relate solely to the method of stating equations from the prob- 

 lems 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-meng in 1299, and his Szii-yiian Yii-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 so- 

 called 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." 



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,^^ 

 written in the fifth century; of the Suan-ching, one of the great trea- 

 tises on Chinese arithmetic ; nor of Chin Lwan who wrote the Wu-king- 

 suan-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 sci- 

 ence 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 prob- 

 ably found out for themselves certain truths of geometry, and among 

 these the Pythagorean theorem ; that they early developed a plane trig- 

 onometry; that they did good work in approximating the value of tt; 

 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 End5, Hayashi, Kikuchi, Fujisawa and Mikami have made of the 

 native Japanese treatises. When this comes to pass we may possibly 



" Li Yay, as Cantor has it. 



" It seems first to have appeared in print in a work by Apianus, 1527. 



" Five sections. 



