=127.= The pure doctrine of extension, as constructed by Grassmann, need not be discussed--it included much empirical material, and was philosophically a failure. But his principles, I think, will enable us to prove that projective Geometry, abstractly interpreted, is the science which he foresaw, and deals with a matter which can be constructed by the pure intellect alone. If this be so, however, it must be observed that projective Geometry, for the moment, is rendered purely hypothetical[132]. All necessary truth, as Bradley has shown, is hypothetical[133], and asserts, _primâ facie_, only the ground on which rests the necessary connection of premisses and conclusion. If we construct a mere conception of externality, and thus abandon our actually given space, the result of our construction, until we return to something actually given, remains without existential import--if there _be_ experienced externality, it asserts, then there must be a form of externality with such and such properties. That there must be experienced externality, Kant's first argument about space proves, I think, to those who admit experience of a world of diverse but interrelated things. But this is a question which belongs to the next Chapter. What we have to do here is, not to discuss whether there is a form of externality, but whether, if there be such a form, it must possess the properties embodied in the axioms of projective Geometry. Now first of all, what do we mean by such a form? =128.= In any world in which perception presents us with various things, with discriminated and differentiated contents, there must be, in perception, at least one "principle of differentiation[134]," an element, that is, by which the things presented are distinguished as various. This element, taken in isolation, and abstracted from the content which it differentiates, we may call a form of externality. That it must, when taken in isolation, appear as a form, and not as a mere diversity of material content, is, I think, fairly obvious. For a diversity of material content cannot be studied apart from that material content; what we wish to study here, on the contrary, is the bare possibility of such diversity, which forms the residuum, as I shall try to prove hereafter[135], when we abstract from any sense-perception all that is distinctive of its particular matter. This possibility, then, this principle of bare diversity, is our form of externality. How far it is necessary to assume such a form, as distinct from interrelated things, I shall consider later on[136]. For the present, since space, as dealt with by Geometry, is certainly a form of this kind, we have only to ask: What properties must such a form, when studied in abstraction, necessarily possess?