In practice, the geometrician will retain the type of ordering relation imposed by our sensory experience and will conceive, by abstraction, of a mathematical space of points which will manifest itself as three-dimensional when this ordering relation is adhered to. There is nothing to prevent him, however, from conceiving of mathematical spaces of any whole number of dimensions, either by modifying the ordering relation or again by modelling his mathematical manifold on some -dimensional sensory continuum. Let us suppose, then, that we have conceived of a three-dimensional mathematical space, obtained as an abstraction from the three-dimensional space of common experience. Just as was the case with our sensory continuum, the mathematical continuum will be amorphous; no intrinsic metrics will be inherent in it, hence it will present us with no definite geometry. The definition of congruence, that is, of the equality of two spatial stretches and more generally of two volumes, and the identity of shapes and sizes, will remain as conventional as before; and it will be only after we have introduced measuring conventions into an otherwise indifferent mathematical space that Metrical Geometry as opposed to Analysis Situs will be possible. However, the discussion of these points will be reserved for the next chapters. [Pg 32] CHAPTER II THE BIRTH OF METRICAL GEOMETRY IN the preceding chapter, we mentioned that the concept of continuity was suggested by our sense experiences; and that our understanding of space as a three-dimensional continuum had arisen from a synthesis of our various sense impressions. In the present chapter, we shall be concerned more especially with the geometry of the space continuum; and we shall show how measurement in turn would appear to have arisen from experience, more particularly visual experience.