Since at least Dai Zhen's time, the procedures contained in Chapter 2 of The Nine Chapters on mathematical procedures (Jiuzhang suanshu) and entitled “Directly lü-ing (jinglü)” have caused difficulties to the philologists, who regularly heavily modified the received text as well as that of the commentaries by Liu Hui and Li Chunfeng on it. One particular problem that these procedures raise is that their description does not exactly fit with the description of the main procedure of Chapter 2, in The Nine chapters, the procedure called “Suppose,” although they clearly depend on it. One of the commentaries stresses this point, whereas the other stresses the connection to the main procedure of the chapter, despite its actual formulation. The paper suggests that the newly excavated Book of mathematical procedures (Suanshushu) casts light on both these procedures and the commentaries. The Book of mathematical procedures contains procedures on the basis of which the procedures “Directly lü-ing (jinglü)” were clearly obtained by abstraction. This hence provides evidence showing why the procedures were described in a way that does not correspond to the procedure for the operation named “Suppose.” Moreover, these sources allow analyzing how abstraction was carried out within the framework of algorithms in ancient China. Lastly, the paper suggests a possible origin for the key concept of lü, which was so important for both The Nine chapters and the commentaries on it
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