Additive and subtractive as relational entities in the algebra of al-Zanjānī (and his predecessors)

In the texts of the arithmetical-algebraic tradition initiated by al-Karajī (11th c.), the terms of a composite algebraic expression are considered to be positive or negative only in the sense of relational entities. Indeed, within the rules for operating with these algebraic expressions, additives and subtractives represent operands, and tabular methods allow one to compute with subtractives more freely. The strategies adopted for impossible problems further confirm the relational nature of these entities. In my article, I examine these aspects by considering the work of al-Zanjānī (13th c.) and of his predecessors al-Karajī and al-Samaw'al.