 An important part of the dichotomy, as traditionally understood, was that magnitudes and ratios of them were not thought of as numerical entities, with arithmetic operations defined on them, but in a more concrete way based on geometry (as lengths, areas, volumes, angles, etc. and as relations between them). More particularly, while Eudoxos’ theory provides a contextual criterion for the equality of ratios, it does not include a definition of the ratios themselves, so that they are not conceived of as independent objects (Stein 1990, Cooke 2005). Such features do little harm with respect to basic applications of the theory; but they lead to inner-mathematical tensions, e.g., when solutions to various algebraic equations are considered (some of which could be represented numerically, others only geometrically). This tension came increasingly to the fore in the early modern period, especially after Descartes’ integration of algebra and geometry. What was called for, then, was a unified treatment of discrete and continuous quantities.