1. These rules seem to have been invented around the 11th century. According to Yang Hui (fl. ca. 1275) they were already used in the Zhinan suanfa (ca. 1078–1189) (cf. Yamazaki (2), 1962, p. 137). According to Zhu Shijie “the classical method (of division) is based on the shangchu technique (i.e. that which uses three superposed rows); but because it is difficult for beginners it has been replaced by the jiugui method which is not orthodox.” (From the Suanxue gimeng, preliminary chapter).
2. These are the rules of the Suanfa tongzong (1592).
3. Example taken from the journal Zhusuan (The Abacus), 1983, no. 3, p. 44.
4. Cf. chapter 16344 of the Yongle dadian encyclopedia (1407) which is devoted to the extraction of roots and contains computational procedures due to numerous arithmeticians.
5. Cf. Wang Ling and Needham (1), 1955, pp. 359 and 362.