Introduction

The present work explores the role and presence of distributivity-like properties in some texts of the medieval Euclidean traditions. It starts with an overview of propositions in Euclid’s Elements which, retrospectively seen, embody treatments of distributivity-related properties of multiplication over addition or subtraction. It examines the way in which, taken together, these propositions and their interrelations are related to the all-important separation in Euclid’s treatise between its arithmetic and geometric parts, and, consequently, between continuous and discrete magnitudes. The following main three sections discuss the treatment of distributivity-related situations in various medieval mathematical treatises, against the background of the significant changes that affected the interrelation between the two domains, geometry and arithmetic. The texts discussed belong to three main medieval mathematical traditions: Islamicate, Latin and Hebrew. They display a wide array of attitudes towards distributivity-like situations: some either explicitly or tacitly rely on the relevant Euclidean propositions, some formulate distributive-like rules in order to attribute them a foundational role or to use them as ad-hoc resources in proofs, and some others simply bypass them even where their use would seem to be helpful. The perspective afforded by examining these distributivity-like situations, in various manifestations and in varying contexts, gives rise to fresh insights concerning medieval attitudes towards the questions of what are numbers and magnitudes, how they are used, what are their basic defining properties, and what is the right way to provide clear foundations for arithmetic as an autonomous field of mathematical knowledge.