Strictly speaking, this book is not about the history of mathematics, except indirectly. Rather, it is devoted to commentaries, a specific textual form that many practitioners of mathematics, like other scholars active in a variety of other domains, have sometimes chosen in order to carry out their inquiries. To the modern reader, commentaries might look like a rather odd phenomenon: but until early modern times, for an author to write a commentary on a base text of any sort was quite a common choice of genre, and the practice has certainly not disappeared in our own age. By “commentary,” here, we do not mean merely the activity of jotting down one’s thoughts for oneself while reading a text or afterwards, but, instead, deciding to compose not only for oneself but above all for other readers (known and unknown students, colleagues, and others) a kind of text that complies with a base text but remains ancillary to it.

In the early decades of the Tang dynasty (618–907), many scholars were involved in official projects whose aim was to produce authoritative editions of canonical texts. As a result, The Ten Canonical Texts of Mathematics and The Proper Meaning of the Five Canonical Texts were presented to the Tang throne, respectively, in 656 and 653, with, in both cases, selected ancient commentaries and subcommentaries composed at the time. In both contexts, there existed mathematical commentaries, and mathematical subcommentaries were composed on them. The aim of this chapter is to analyze and contrast different scholars’ mathematical commentaries and subcommentaries on both the mathematical and Confucian canons. The authors show that in the framework of both projects, exegetes aimed at highlighting the meaning/intention (yi) of the text commented upon. More importantly, mathematical commentaries and subcommentaries allow the authors to show that far from being a loose concept, yi referred to a quite precise notion of “meaning.” However, interestingly, this notion of “meaning,” and the part of the text commented upon in which exegetes located it, show striking differences, depending on whether exegetes operated in the context of the Confucian canon or in that of the mathematical canon. On the one hand, in the context of the mathematical canon, Liu Hui (fl. 263) and Li Chunfeng (602–670) both focused on looking for the meaning/intention yi of the procedures. In the commentary and subcommentary on The Nine Chapters on Mathematical Procedures, in order to fulfill this task, the exegetes relied on diagrams (tu), blocks (qi), and problems (wen), while, in his subcommentary on Zhen Luan’s (fl. sixth century) commentary on Mathematical Procedures of the Five Canonical Texts, Li Chunfeng added problems and reformulated procedures to reveal the yi of Zhen Luan’s procedures, where he believed Zhen Luan failed to do it. Conversely, in the context of the Confucian canon, although Kong Yingda (574–648) and Jia Gongyan (fl. 650–655) also looked for the yi, for them, the yi on which they focused was located in quantities stated by previous commentators (such as Zheng Xuan, 127–200). Additionally, the two sets of exegetes did not practice mathematics in the same way. For instance, by contrast with exegetes who composed commentaries for the mathematical canon, Jia and Kong did not rely on any computational tool. The authors conclude that, although scholars’ aims were partly the same in the two contexts, their understandings of, as well as their commentarial and mathematical practices in relation to, the yi differed. This chapter also explains why the differences established between the mathematical commentaries and subcommentaries in each of the two contexts could also provide promising tools to distinguish the two layers of texts where they were blurred.

This chapter is devoted to what I call textual anachronism. By this expression, I refer to forms of anachronism that lead to interpreting ancient texts on the basis of anachronistic assumptions with respect to how these texts made sense for the ancient actors. This is, for instance, the case when historians take the textual components that they find in ancient documents (like a mathematical problem, an algorithm, a proof, and a diagram), as they would take what they consider to be modern counterparts, and interpret these components on that basis. I argue that this form of anachronism has caused misinterpretations of several kinds, which I identify and analyze, using examples from four historical contexts. I further discuss the historiographical implications of this type of anachronism. I conclude with the thesis that one can limit the effects of this form of anachronism by using a historical approach to forms of text. Moreover, a historical approach of this kind sheds light on the fact that anachronism has a history that might also be interesting to consider.