Reading and (Re-)​Classifying Canonical Instructions of the Past: Commentaries on ‘The Nine Chapters on Mathematical Procedures’ from the 3rd to the 13th Centuries

The earliest extant Chinese mathematical writings include two types of components of particular interest for our discussion on manuals and handbooks. On the one hand, there are mathematical problems that often evoke tasks carried out by officials working in the imperial bureaucracy. On the other hand, there are mathematical “procedures,” or “algorithms” in today’s parlance, to solve such problems. This description fits most of the mathematical books composed in China until the seventh century.

Since the nineteenth century, historians have regularly formulated the view that these writings were instructional manuals for officials, or more broadly for people learning mathematics. The important term here is “instructional.” Indeed, for these modern scholars, “procedures” (or “rules”) were only means to ends. According to the logic of their interpretation, officials must have used these books in pieces, extracting procedures as necessary to achieve certain goals. In fact, historians have regularly ascribed practicality to the goals that for them the mathematical problems exemplified. In their view, texts of “procedures” were meant to be “obeyed,” not understood. Following the rules required no knowledge, certainly no mathematical knowledge as such.¹ These historiographical assumptions, in turn, have often informed how historians have read these ancient algorithms. In line with such interpretations, mathematical sources were understood to evidence a divide between “East” and “West.”

This is the point where historiographies of handbooks and manuals meet with historiographies of mathematics in the ancient world. The interpretations just outlined read mathematical sources composed of problems and algorithms as handbooks or manuals. They make a priori assumptions about the type of knowledge handbooks and manuals require from and convey to their readers. Accordingly, they often consider handbooks and manuals to represent lower forms of scientific activity. In brief, at the root of their conception lies a specific representation of the meaning and value of instructional texts, especially in relation to mathematical practice.

Our conference invites us to suspend any a priori and uniform judgment about handbooks and manuals, to look instead at writings providing the means to do things on a case by case basis. If we accept the view that the earliest extant Chinese mathematical writings provided (at least partly) the means to do things in the form of “procedures,” what can these writings tell us about the knowledge required to produce a collection of such “procedures” and the knowledge needed to use them?

I address these issues, focusing on The Nine Chapters on Mathematical procedures (Figure 1), which in my view was completed in the first century CE.² Why this book?

First, it shows that complex operations were required to produce instructional texts. I illustrate this point with procedures for the so-called “excess and deficit” and the like; procedures for multiplying quantities with integers and fractions; and procedures for dividing a quantity of this kind by another like quantity. In The Nine Chapters, the related procedural texts show that mathematical knowledge was needed not only to determine which arithmetical operations one should use to solve any possible case but also to write up a single text of “procedure” that covered all possible cases. In fact, the textual properties of these texts reveal that actors prized the epistemological value of generality in specific ways. These textual properties reveal that actors possessed mathematical knowledge meaningful for a history of algebra and a history of algorithms. They further highlight competences readers had to acquire to use these instructional texts. Readers had to know how to execute mathematical operations, and also how to interpret and employ texts of procedures. Elsewhere I argue that, to turn the texts into action, readers had to understand the “meaning” of the operations prescribed.³ This exposes the limits of the divide between following a rule and understanding it.

My second reason for concentrating on The Nine Chapters is that the book groups procedures into chapters. It thus shows the knowledge involved in classifying procedures.

Third, The Nine Chapters is the only ancient Chinese mathematical book composed of problems and procedures for which we can document different readings. Indeed, the book was handed down with two layers of commentary from the third and the seventh centuries. All ancient editions of The Nine Chapters include these commentaries, which I designate as “the ancient layers of commentary” since it is difficult to distinguish them from one another. Moreover, in the eleventh and thirteenth centuries, scholars resumed commenting on The Nine Chapters and its two ancient layers of commentary. Two witnesses for this are Jia Xian’s 賈憲 eleventh-century and Yang Hui’s thirteenth-century subcommentaries, which were handed down only partially. We can read them in Yang Hui’s Mathematical Methods Explaining in Detail the Nine Chapters, completed in 1261, in which they appear combined in a way that is not yet clear. I thus designate them “the later layers of commentary.” Only some chapters of Yang Hui’s Mathematical Methods came down to us, in Song Jingchang’s 1842 critical edition (Figure 2).⁴ The only one of these commentators about whose life we have substantial evidence, beyond the fact that he produced his writings, is Li Chunfeng, who was active in the seventh century at court and wrote about mathematics, astral sciences, and the history of scholarship.

The essential point is this: the ancient and later layers of commentary attest to how readers of the past interpreted, and handled, ancient instructional texts from The Nine Chapters. Consequently, instead of projecting modern engagements with texts onto ancient sources, as historians have done since the nineteenth century, we have pieces of evidence with which to approach—in a historical way—how actors of the past dealt with old texts of “procedures.” This is the method I use to problematize and historicize what it meant, in these contexts, to practice and write up mathematics in the form of problems and procedures.

The later layers of commentary rename and rewrite procedures of The Nine Chapters. Furthermore, in a separate chapter of Mathematical Methods, Yang Hui reclassifies the procedures of the ancient canon using new names and formulations. These practices provide evidence for examining what, from an actor’s viewpoint, was at stake in the two operations of naming and classifying.

My strategy to address the issues at stake looks like this:

I mainly concentrate on Chapter 7 of The Nine Chapters, titled “Excess and Deficit,” which begins with the eponymous procedure. To determine the unknown(s) sought in the context of a given mathematical problem, the procedure relies on four numbers: two suppositions made on a given magnitude, and the excess and deficit these suppositions respectively cause with respect to another given quantity. Apparently, the text of this procedure caused its readers problems because the ancient editions of The Nine Chapters and the ancient layers of commentary present presumably more variations here than anywhere else. The editorial interventions of modern editors are also the most significant.

In my view, the key difficulty was this: the text of the procedure begins with two lists of operations (I call them I and II), whose combination determines the value of the magnitude on which the suppositions were made. This is the unknown for some problems in Chapter 7. The text of the procedure goes on with a conditional and a third list of operations (III). This list uses the results of list I (jumping over list II) to determine the unknown quantities that bring suppositions, excess and deficit, into relation. These are the unknowns for the other problems in Chapter 7. The procedure (and the similar procedures of, respectively, two excesses, two deficits, etc.) is what lends the chapter coherence. The first eight problems of Chapter 7 to which the conditional refers are solved by lists I and then III, whereas the twelve other problems are solved by lists I and then II.

This feature illustrates the complex operations carried out in producing the instructional text. Since the solutions of the two types of problems shared a common list of operations (I), a single procedure, with a specific structure, was shaped for both, even though the unknowns sought-for are structurally different, and the meaning of the operations is accordingly different. Producing this text of procedure involved mathematical knowledge of an algebraic kind. It shows that actors did not use procedures as independent pieces, but inquired into connections between them. This chimes with a way of valuing generality in this context, for which we have more evidence.

To use such procedural texts to do things, the reader must be aware that the text requires a specific form of navigation to yield the correct list of operations. Only then can the reader extract lists I and II for some problems, and lists I and III for others.

The older layers of commentary attend to the “meaning” of the successive operations, establishing why the procedure is correct in all the cases for which it is used. They also make clear how some prescriptions should translate into action. Note that these commentaries do not discuss the goals of these applications as practical, but as requiring different types of mathematical knowledge. Readers thus seem to look for procedures that can be applied to different mathematical cases, not different concrete cases.

We can rely on the older layers of commentary to approach the knowledge required to use these texts of procedures. The later layers of commentary also shed light on this issue, but in another way. In the later layers, the commentary takes the form of reformulating the “procedure” of the canon as a “method.” The reformulation—and the annotations added between the steps of the “method”—show other facets of the knowledge required to read and handle the text because the commentary now feels the need to provide further explanation on how to turn the text into action. Interestingly, these reformulations and annotations are what later books claiming to rely on The Nine Chapters take for the ancient canon. This is the case, for instance, with Wu Jing’s 吳敬 All-Encompassing Compendium of the Detailed Annotations on The Nine Chapters and Analogical (Problems) (Figure 3). This insight yields essential information for extracting evidence from the All-Encompassing Compendium that can shed light on parts of the later, long since lost layers of commentary.

The later layers of commentary for Chapter 7 show that Yang Hui did not grasp how to handle the text of the procedure. This conclusion is confirmed by Yang Hui’s rewriting of other texts of procedure that required similar handling. He did not understand that the text yields lists I and II for some problems, but lists I and III for others. Consequently, he considered the whole procedure (lists I, II, and III, one after the other) was needed to solve the first type of problems.

Accordingly, for the second type of problems in Chapter 7, Yang Hui provides alternative methods. In the chapter “Compiling the Categories,” in which he compiles his “methods” to reclassify the problems of The Nine Chapters, he offers a reorganization of the ancient chapter.

This effort reveals the key relationship between formulating procedures and classifying problems. It also shows the knowledge involved in such formulation and classification.