Therefore we shall now try to clarify the nature of these things that we call buildings by the following brief consideration.
For one thing, what is the relation between location and space? For another, what is the relation between man and space? The bridge is a location. As such a thing, it allows a space into which earth and heaven, divinities and mortals are admitted. The space allowed by the bridge contains many places variously near or far from the bridge. These places, however, may be treated as mere positions between which there lies a measurable distance; a distance, in Greek στάδιον, always has room made for it, and indeed by bare positions. The space that is thus made by positions is space of a peculiar sort. As distance or Stadion, it is what the same word, Stadion, means in Latin, a spatium, an intervening space or interval. Thus nearness and remoteness between men and things can become mere intervals of intervening space. In a space that is represented purely as spatium, the bridge now appears as a mere something at some position, which can be occupied at any time by something else or replaced by a mere marker. What is more, the mere dimensions of height, breadth, and depth can be abstracted from space as intervals. What is so abstracted we represent as the pure manifold of the three dimensions. Yet the room made by this manifold is also no longer determined by distances; it is no longer a spatium, but now no more than extensio—extension. But from a space as extensio a further abstraction can be made, to analytic-algebraic relations. What these relations make room for is the possibility of the construction of manifolds with an arbitrary number of dimensions. The space provided for in this mathematical manner may be called "space," the "one" space as such. But in this sense "the" space , "space," contains no spaces and no places. We never find in it any locations, that is, things of the kind the bridge is. As against that, however, in the spaces provided for by locations there is always space as interval, and in this interval in turn there is space as pure extension.