>many problems are not solved by algebra, but by a method I call “arithmetical reasoning”. Here one merely reasons through the stated operations to find the unk

>many problems are not solved by algebra, but by a method I call “arithmetical reasoning”. Here one merely reasons through the stated operations to find the unknown. No unknown is named, and no equation is set up. The presence of these non-algebrac solutions suggest that al-Khwārizmī did not write all the problems himself. He seems to have taken over a corpus from an oral tradition. Questions: 1) What did this pre-algebraic reasoning actually look like? 2) Was is al-Khwarizmi that elevated problem solving to a more mechanical process or is it inappropriate to credit one person with that shift? 3) When did Europeans start using algebra (e.g. with named variables and rules for manipulating them)? 4) Did al-Khwarizmi ever think about the algorithmic efficiency of the procedures he proposed or used? For example, is there ever an explicit preference given for a more efficient procedure (in terms of the formula size or magnitude of a number)?