Already in the Rules for the Direction of the Mind (1628) Descartes formulated his fundamental theory of extension: “By extension we mean whatever has length, breadth and depth” (AT X 442, CSM I 58). These three dimensions are inseparable in reality: a surface or line is not really distinct from extension but only considered so by an abstraction of the intellect (AT X 446, CSM I 61). And the difference between them is merely “nominal” (AT X 449, CSM I 63). We can speak of countless other measurable “dimensions” of an extended thing, such as speed or number of parts. But these quantifiable modes of extension “add absolutely nothing to the things which possess them” (AT X 448, CSM I 62). Simply put, extension is three-dimensionality.

Anything extended is a body. Indeed, we should not say “body possesses extension” since we do not have different conceptions of body and extension: “We might just as well say … ‘that which is extended is extended’” (AT X 444, CSM I 60). In The World (ca. 1633), Descartes writes that extension is not a mere accident of body but “its true form and essence” (AT XI 36, CSM I 92). Later, in the Principles of Philosophy (1644), he formulates the same conception using the technical notion of attribute. Extension is the “principal attribute” of body: “Extension in length, breadth and depth constitutes the nature of corporeal substance” (AT VIIIA 25, CSM I 210). Unlike variable modes, attributes are merely “conceptually distinct” from the substance itself: we can have a clear and distinct idea of body apart from a certain shape or motion, but not apart from extension. The principal attributes thought and extension are “nothing else but thinking substance itself and extended substance itself, that is, mind and body” (AT VIIIA 30–31, CSM I 215). Furthermore, there is merely a conceptual distinction among the various attributes of a given substance. Thus, Descartes mentions that a body's divisibility into parts (AT IXB 53, CSM I 215 n. 1) and quantity (AT VIIIA 44, CSM I 226) are merely conceptually distinct from its extension (see distinction [real, modal, and rational]). This identification of body, extension, quantity, and divisibility is a crucial step in Descartes’ program to mathematize natural philosophy: “I recognize no matter in corporeal things apart from that which the geometers call quantity” (AT VIIIA 78, CSM I 247).