Similarities between Chinese and Arabic Mathematical Writings: (I) Root extraction

Abstract

First century Chinese, fifth century Indian, and Arabic documents from the 9th century onwards, contain similar tabular procedures to extract square and cube roots on place-value numeration systems. Moreover, an 11th century Chinese astronomer, Jia Xian, as well as al-Samaw'al, a 12th century Arab mathematician, extracted roots of higher order with the so-called Ruffini-Horner procedure. This article attempts to define a textual method to organize this corpus, by distinguishing relevant criteria for identifying similarities and differences from a historical as well as conceptual point of view. The first part analyses three different states of the descriptions of algorithms in China between the 1st and the 11th centuries, all of which exhibit a definite historical stability. The rewriting which allows one to proceed progressively from one state to the next shows a uniformity in the components of the algorithm, which culminates in procedures of the type Ruffini-Horner. Textual criteria demonstrate a greater affinity of certain algorithms, such as those described by Kūshyār ibn Labbān (ca 1000) with Chinese rather than with Indian texts, which are in turn closer to algorithms described by al-Khwārizmī. Criteria of the same kind link the algorithms of Jia Xian and al-Samaw'al on the one hand, and those of Kūshyār and al-Samaw'al on the other.