This article starts with an observation. The chapters that compose the earliest extant mathematical book handed down through the written tradition in China, The Nine Chapters on Mathematical Procedures, and that are mentioned in the title of the book, are a key feature distinguishing this book from the manuscripts excavated from tombs sealed in China during, roughly speaking, the Qin and early Western Han dynasties. We argue that as early as the Eastern Han dynasty, the names of these chapters were related to an interpretation of an expression ("the nine parts of mathematics") occurring in a text essential for Confucian scholars, Zhou Rites. We thus suggest that these facts allow us to make sense of both the third century commentator Liu Hui's account of the history of The Nine Chapters, and the difference between The Nine Chapters and the newly discovered manuscripts with respect to the textual parts composing them. We argue that in Liu Hui's view, the fact that The Nine Chapters was organized into such chapters was a feature manifesting the adherence of this mathematical book to a set of texts to which the first Qin emperor had gone to great lengths to control the access. This hypothesis allows us to interpret Liu Hui's assertion that the transmission of The Nine Chapters was badly affected by the first Qin emperor's book burning policy, at the same time as it accounts for why the newly excavated documents are not divided into chapters of this kind. These conclusions suggest that the chapters, and their names, might have played a key role in The Nine Chapters, these textual parts constituting an essential feature distinguishing this Classic from other mathematical writings. In the second part of our argument, we discuss a mathematical interpretation of the meaning of these chapters and the chapterization.