Page:Popular Science Monthly Volume 80.djvu/603

Rh 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) 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 (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 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, 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