According to the history of the Song Dynasty, Xian was a palace eunuch 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 invented the Jia Xian triangle around the first half of the 11th century, about six centuries before Pascal. Jia Xian used it as a tool for extracting square and cubic roots. The original book by Jia Xian titled Shi Suo Suan Shu was lost; however, Jia's method was expounded in detail by Yang Hui, who explicitly acknowledged his source: "My method of finding square and cubic roots was based on the Jia Xian method in Shi Suo Suan Shu." A page from the Yongle Encyclopedia preserved this historic fact.
Jia Xian's additive-multiplicative method implemented the "Horner" rule.
Additive-multiplicative method of square-root extraction
Additive-multiplicative method of cubic-root extraction
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