Jia Xian triangle (Pascal's triangle) usingrod numerals, as depicted in a publication ofZhu Shijie in 1303 AD.Yang Hui referred to Jia Xian's Shi Suo Suan Shu in the Yongle Encyclopedia
According to the history of the Song dynasty, Jia was a palaceeunuch of the Left Duty Group. He studied under the mathematician Chu Yan, and was well versed in mathematics, writing many books on the subject. Jia Xian described thePascal's triangle (Jia Xian triangle) around the middle of the 11th century, about six centuries beforePascal. Jia used it as a tool for extractingsquare andcubic roots. The original book by Jia entitledShi Suo Suan Shu was lost; however, Jia's method was expounded in detail byYang Hui, who explicitly acknowledged his source: "My method of finding square and cubic roots was based on the Jia Xian method inShi Suo Suan Shu."[1] A page from theYongle Encyclopedia preserved this historic fact.
Jia Xian's additive-multiplicative method implemented the "Horner" rule.[2]
Additive-multiplicative method of square-root extraction
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S Guo, Preliminary research into Jia Xian's Huangdi Jiuzhang Suanjing Xicao (Chinese), Studies in the History of Natural Sciences 7 (4) (1988), 328 -334.
S Guo, Jia Xian, in Du Shiran (ed.), Zhongguo Gudai Kexuejia Zhuanji (Biographies of Ancient Chinese Scientists) (Beijing, 1992), 472 -479.
R Mei, Jia Xian's additive-multiplicative method for the extraction of roots (Chinese), Studies in the History of Natural Sciences 8 (1) (1989), 1 -8.
