The Siyuan yujian 四元玉鑑 (Jade Mirror of Four Unknowns, 1303), written by Zhu Shijie 朱世傑 (13th – 14th c.), is a text which discusses an algebraic method with up to four unknowns, based on the procedure of the celestial element (tianyuan shu 天元術), the place-value notation for polynomials and equations developed in China’s Song Dynasty (960 – 1279). The procedure of the celestial element was lost in China since the middle of the 14th century, and it was rediscovered in the 18th century. This method might have been transmitted to Korea in the late 13th century, but there is no evidence to suggest that the Siyuan yujian was also brought to Korea before the 19th century. Unlike in China, there seems to be a continuous tradition of Korean mathematics in the practice of the procedure of the celestial element, and when the calculation by borrowed root and powers (jiegenfang 借根方), the cossic algebra taught to emperor Kangxi 康熙 (r. 1662 – 1722) by Antoine Thomas, was transmitted to Korea, Korean mathematicians seemed to have learned the method based on their understanding of the procedure of the celestial element. The Siyuan yujian was republished in China in the early nineteenth century, but the text itself was difficult to understand. Luo Shilin 羅士琳 (1789 – 1853) provided calculation processes for each problem in his Siyuan yujian xicao 四元玉鑑細艸 (Detailed Calculations for the Jade Mirror of Four Unknowns), which was soon brought to Korea, and for this work the Korean mathematician Nam Pyŏng-Gil 南秉吉 (1820-1869) wrote a commentary, the Oggam secho sanghae 玉鑑細艸詳解 (Comprehensive Explanations of the Detailed Calculations for the Jade Mirror, 1850s). In this text, Nam provided some procedures different from those of Luo’s, one of which is that he tried to avoid negative powers in order to make simpler the counting rod expressions. Another interesting piece of commentary tells that he used the idea of “equality” in the procedures of polynomial addition and root extraction, which suggests influence from the calculation by borrowed root and powers. Later in another work of his, he explains counting rod expressions with ideas of “row” and “column”, and he seemed to have changed his mind on his opinions about the procedure of the celestial element and the calculation by borrowed root and powers.