ESHS/HSS Annual Meeting

Diagrammatizing computations in Chinese and Arabic sources: From decimal place-value numeration systems to place-value notations for polynomials

• In Event: Diagrammatic Computations and Their Symbolic Aspects (I)

English Abstract

The work Sea-Mirror of the Circle Measurements (Ceyuan haijing 測圓海鏡), which Li Ye 李冶 (1192-1279) completed in 1248, attests to the use of polynomial computations to establish algebraic equations solving problems. More precisely, Li Ye prescribes computations with polynomials written using a vertical place-value notation, their coefficients being numbers written using a horizontal place-value notation. The author states the operands and the results of these computations, without making explicit how the operations with polynomials were executed. Elsewhere, I have argued that the two parts of the solution presented by Li Ye for each problem are correlated, and that this correlation enables us to put forward a hypothesis on how some of these operations were executed. My talk aims to establish that these computations with polynomials were diagrammatized in a way symmetrical to how computations with numbers written with a decimal place-value system were. I will then compare this way of diagrammatizing computations with those attested to in two Arabic sources: computations with sexagesimal place-value notations presented in Kushyar ibn Labban’s 11th-century Principles of Hindu Reckoning and computations with polynomials presented in As-Samaw’al’s 12th-century The Bright (al-Bāhir). In all these cases, we have the articulation of two place-value notations. Moreover, these books all attest to a way of correlating the diagrammatization of computations with numbers and that of computations with notations articulating two place-value notations. My conclusion aims at highlighting the intimate and recurring relation between computations that, in the historiography, were perceived as belonging to the different contexts of arithmetic and algebra.

Author

• Karine C. C. Chemla, University of Edinburgh and SPHERE, CNRS-UPC