Mathematical Remarks on a List of Problems Attributed to Ibn al-Khaww¯am

Abstract

History of mathematics is laden with problems which are difficult to understand andinterpret from linguistic point of view and it is hard to represent faithfully medieval orancient problems with modern notations without committing mistakes. It is only natural towant to improve the quality of material in history of mathematics by writing a corrigendum.In Derouiche’s list of problems attributed to Ibn al-Khaww¯am, several conditions that anexpression be a square are replaced by equations setting it equal to zero. Proposed solutionssometimes contain errors or are incomplete. In Problem 6, the elimination of a squareroot changes the equation and excludes an elementary solution of the original system. InProblem 28, a correct reduction to a cubic is followed by an incorrect account of its real roots:the original equation has two positive real solutions, albeit neither is rational. The calculationsbelow establish these distinctions in detail, give corrected derivations, and prove the rationalimpossibility of the reconstructed Problem 11 by reduction modulo seven. Where the Englishwording and the displayed equations admit different interpretations, the alternatives areexamined separately. Their mathematical consequences can be decided, while the historicalreading still requires comparison with the original source