According to the history of the Song dynasty, Jia 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 described the
Pascal's triangle (Jia Xian triangle) around the middle of the 11th century, about six centuries before
Pascal. Jia used it as a tool for extracting
square and
cubic roots. The original book by Jia entitled
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==