Calculations of Al-Khwarizmi -- sources of algebra and arithmetic. (Q2856939)

The paper analyses the two famous treatises by Al-Khwarizmi -- the arithmetical treatise (only in Latin translations, usually called \textit{Algorismo de indo numerorum}) and the earlier algebraic treatise (\textit{Kitab al-hisab al-jabr w'al-muqabalah}), based on their recent translations into Czech by P. Vopěnka. The author presents major parts of both works and focuses on their two distinct features: the operative use of the Hindu numerals, and Al-Khwarizmi's algorithms for equations. The author also compares Al-Khwarizmi's proof of the Pythagorean theorem to the corresponding section in Euclid's \textit{Elements}.NEWLINENEWLINE\textit{Ladislav Kvasz}'s theory of mathematical languages [Patterns of change. Linguistic innovations in the development of classical mathematics. Basel: Birkhäuser (2008; Zbl 1153.01002)] is used throughout the paper in discussing the level of advancement of Al-Khwarizmi's arithmetic and geometric algebra in comparison to earlier Graeco-Roman traditions. The author also frequently refers to \textit{D. Knuth}'s assessment of Al-Khwarizmi's algorithms [Mater. Int. Symp., Urgench/UzSSR, 64--98 (1982; Zbl 0583.68006)]. The author highlights the potential of Hindu numerals as used by Al-Khwarizmi to express any natural and positive rational number, and the facility of basic operations conducted with it. She also draws attention to the expressive power of Al-Khwarizmi's algorithms (albeit only rhetorically formulated) in contrast to geometry (e.g., expression of higher powers than three, demonstration of the non-existence of solutions).NEWLINENEWLINEFor the entire collection see [Zbl 1330.01007].