CHAPTER IV - DESCRIPTIVE GEOMETRY

Summary

Application To Descriptive Geometry. An extensive manifold of ν - 1 dimensions is a positional manifold of ν - 1 dimensions with other properties superadded. These farther properties have in general no meaning for a positional manifold merely as such. But yet it is often possible conveniently to prove properties of all positional manifolds by reasoning which introduces the special extensive properties of extensive manifolds. This is due to the fact that the calculus of extension and some of the properties of extensive manifolds are capable of a partial interpretation which construes them merely as directions to form ‘constructions’ in a positional manifold. Ideally a construction is merely an act of fixing attention upon a certain aggregate of elements so as to mark them out in the mind apart from all others; physically, it represents some operation which makes the constructed objects evident to the senses. Now an extensive magnitude of any order, say the ρth, may be interpreted as simply representing the fact of the construction of the subregion of ρ - 1 dimensions which it represents. This interpretation leaves unnoticed that congruent products may differ by a numerical factor, and that, therefore, extensive magnitudes must be conceived as capable of various intensities.