Non-linear Structures

In Ancient Chinese Mathematical and Cosmographical Treatises
Tuesday
2:00 pm – 3:45 pm
Room B

  • Alexei Volkov, “Textual Structures in Ancient Chinese Mathematical Treatises: On textual Parallelisms, Analogical Reasoning and Didactical Variables”
  • Vera Dorofeeva-Lichtmann, “From Cardinally-Oriented Texts to Textual Interpolations in Traditional Chinese Maps”
  • Karine Chemla, “The Nine Chapters on Mathematical Procedures 九章算術: A Formal Structure with a Cosmological Meaning?”

The main goal of this panel is to discuss non-linear textual structures in Chinese sources relevant to the history of science, in particular, in mathematical and cosmographical texts. The contributions of Volkov and Chemla deal with the structures of early Chinese mathematical texts, while the paper of Dorofeeva-Lichtmann is focused on the early Chinese cosmographic descriptions. This approach to these two categories of texts, even though representing two markedly different fields of proto-scientific expertise, reveal a number of common features worth a thorough investigation. Chemla suggests that the reading of the Han dynasty mathematical treatise Jiu zhang suan shu 九章算術 by its commentators Liu Hui 劉徽 (fl. AD 263) aimed at the identification of structural elements of the treatise which, according to him, may have been lost in the Han 漢 dynasty edition of the treatise. In turn, in his paper, Volkov discusses an earlier study of the Jiu zhang suan shu by the Russian sinologist V.S. Spirin (1929–2002) who identified a non-linear structure in chapter nine of the treatise. Volkov evaluates Spririn’s reconstruction and identifies other non-linear structures in this text. In her paper, Dorofeeva-Lichtmann suggests that there existed a direct link between the early Chinese cardinally-oriented textual structures and textual interpolations found in the extant traditional Chinese maps of Imperial China from the 12th century onwards.

Alexei Volkov, “Textual Structures in Ancient Chinese Mathematical Treatises: On textual Parallelisms, Analogical Reasoning and Didactical Variables”

The paper will begin with a critical evaluation of V.S. Spirin’s (1929–2002) reconstruction of a “nine-term structure” that he discovered in the ninth chapter of the ancient Chinese mathematical treatise Jiu zhang suan shu 九章算術. Spirin’s analysis was published in 1976 in his monograph The Structure of Ancient Chinese Texts (published in Russian) and remained practically unknown to historians of Chinese mathematics. I will critically evaluate Spirin’s analysis and discuss the structures that can be identified in this and other Chinese mathematical treatises. In particular, I will focus on the two following types of structures. The first type is represented by a series of mathematical problems related to the computation of the areas and volumes described in chapters one, four and five of the Jiu zhang suan shu. In this part, I will discuss the interrelationships between the methods that may have been generated by analogical transfer from two-dimensional to three-dimensional case and will explain how the analogical reasoning of this kind may have been interrelated with the structure of this mathematical texts. The second type of structures is related to the didactical dimension of the mathematical treatises; I will show how certain parameters arguably used as didactical variables determined the structure of sequences of problems found in the Jiu zhang suan shu and in other pre-modern Chinese mathematical treatises as well as in later Vietnamese mathematical texts.


Vera Dorofeeva-Lichtmann, “From Cardinally-Oriented Texts to Textual Interpolations in Traditional Chinese Maps”

The formative period of the Chinese Empire—roughly from the late Warring States Period (475/403–222 BC) to the dawn of the Han dynasty (202/206 BC–AD 220)— determined many fundamental principles of the imperial system, including the conceptions of space. The need of the emerging Empire to represent its territory in order to position itself in the inhabited world and cosmos generated a large series of descriptions of terrestrial space. Yet, no cartographical representations of the entire imperial realm have survived from this time, usually explained by the “loss” of early maps.
Earlier (2004, 2007) I advanced a hypothesis that some early descriptions of terrestrial space were structured as cosmograms, which combined properties of a terrestrial description and a cardinally-oriented survey scheme, thus making their (carto)graphical representations (tu 圖) redundant.
The earliest survived general maps of the imperial realm date from the 12th century. These maps were drawn as part of commentarial tradition on early texts and marked a distinction between descriptions of terrestrial space and survey map. At the same time, until the early 20th century, traditional Chinese maps continued to include textual interpolations as cartographic elements.
In the proposed paper I shall try to show a direct link between the early Chinese cardinally-oriented textual structures and textual interpolations in traditional Chinese maps.

Karine Chemla, “The Nine Chapters on Mathematical Procedures 九章算術: A Formal Structure with a Cosmological Meaning?”

The Nine Chapters was completed, in my view, in the first century. Several commentaries on it were handed down, two of them featuring in all the ancient editions. They are the commentary completed by Liu Hui in 263, and the sub-commentary presented by Li Chunfeng to the throne in 656. The commentators made declarations about mathematics or about The Nine Chapters. I have argued that we can interpret these declarations as stating that the two opposed but complementary operations of multiplication and division played a fundamental role in The Nine Chapters or, more generally, in mathematics, which The Nine Chapters displayed since, for the exegetes, it embodied the whole of mathematics. As its title makes clear, the work is composed of nine chapters. My talk draws on this former result to suggest that the earliest known commentaries read a structure in The Nine Chapters that opposed two parts. The first part displayed mathematical patterns of combination of multiplication and division, while the second featured the same patterns repeated. This reading sheds unexpected light on the addition to the canon that Liu Hui composed in the context of his commentary. Indeed, for him, The Nine Chapters as transmitted failed to restore the original canon destroyed by the Qin books burning. His addition, which purports to rely on ancient documents, can be read as completing the canon with respect to the structure described.

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Room B
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In Ancient Chinese Mathematical and Cosmographical Treatises