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Similarities between Chinese and Arabic Mathematical Writings: (I) Root extraction1

Published online by Cambridge University Press:  24 October 2008

Karine Chemla
Affiliation:
Équipe R.E.H.S.E.I.S., C.N.R.S., 27 rue Damesme, 75013 Paris, France Pour Mathilde, ces perspectives internationales

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.

Les documents chinois, depuis le Ier siècle, indiens, depuis le Ve siècle, et arabes, depuis le IXe siècle, contiennent des procédures tabulaires similaires pour l'extraction de racines carrées et cubiques avec des systèmes de numération positionnels. Par ailleurs tant Jia Xian, astronome chinois du XIe siècle, qu'al-Samaw'al, mathématicien arabe du XIIe siècle, ont extrait des racines de degré plus élevé par la procédure dite de Ruffini-Horner. L'article tente de définir une méthode textuelle pour organiser ce corpus, en y distinguant des axes pertinents qui permettent de dégager similarités et différences, d'un point de vue tant historique que conceptuel. Une première partie analyse trois états différents des descriptions d'algorithmes entre le Ier siècle et le XIe siècle en Chine, qui présentent chacun une stabilité historique certaine. La réécriture qui fait passer d'un état au suivant laisse émerger progressivement une uniformité dans les composantes de l'algorithme, laquelle culmine avec des procédures du type Ruffini-Horner. Des critères textuels font apparaître une affinité plus grande de certains algorithmes, tels ceux décrits par Kūshyār ibn Labbān (ca 1000), avec les textes chinois qu'avec les textes indiens, plus proches, eux, des algorithmes décrits par al-Khwārizmī. Des critères de même nature lient, d'une part, les algorithmes de Jia Xian et d'al-Samaw'al, d'autre part les algorithmes de Kūshyār et d'al-Samaw'al.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1994

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