My dissertation aims to challenge the standard narrative in post-Kantian histories of philosophy and science about the 17th century shift from Aristotelianism to mechanism as involving a fundamental reconceptualization of continuous quantity, or extension, and material substance and as creating a deep divide between the ‘modern’ and ‘medieval’ periods. On that view, René Descartes’ understanding of extension as the nature of material substance is seen as one of the hallmarks of that shift. I show that, in fact, (a) late medieval philosophers did not hold one uniform theory of extension but a rich, wide array of views; and (b) a deeper continuity underpins Descartes’ metaphysics and the views of (some) late medieval philosophers. To this end, I examine in detail the views of three representative 14th century thinkers: William Ockham, John Buridan, and Nicole Oresme; and the implications of their views for Descartes’ metaphysics. For each of these authors, I aim to answer the following questions: (1) what kind of entity is extension? (2) Is extension accidental or essential to material substance, using ‘accidental’ and ‘essential’ in this sense: is extension a feature that material substances can, at least by divine power, lack? I investigate these philosophers’ arguments in connection with two phenomena, one natural and the other supernatural: (i) condensation and rarefaction (C/R); and (ii) transubstantiation (T), the Catholic interpretation of the sacrament of the Eucharist. First, on the Aristotelian understanding of C/R, when a body condenses or rarefies, only its extension changes while everything else (prime matter, substantial form, and qualities) stays the same. Thus, C/R was considered a good test case for examining the ontological status of quantity and its relation to material substance. Second, according to T, when the priest blesses the host, the body of Christ takes the place of the bread and comes to be really present on the altar. Yet the qualities of the bread, including its extension, remain without inhering in the substance of the bread. As a result, T has far-reaching implications for the metaphysics of bodies and their accidents. Thus, my investigation interestingly brings together issues in late medieval metaphysics, physics, and theology, issues which by their very nature expose a thinker’s commitments on critical questions about the nature of quantity.