General discussion of the problems of this chapter. The preceding chapter has shewn how, by the recognition of the order of the real points of a line, and by adjunction of postulated points defined by constructions in a limited accessible space, we can build up a geometry in which, with a definite assumption as to the points existing upon a line, Pappus' theorem can be proved. The Real Geometry so arising may then be regarded as a particular case of the Abstract Geometry considered in Chapter I; it was seen that the algebraic symbols which arise in this Real Geometry are precisely similar, in their mutual relations, to the numbers of real arithmetic.

The present chapter is a continuation of Chapter I. It was there shewn that upon the Propositions of Incidence, and the assumption of Pappus' theorem, a theory of related ranges can be built, the correspondence of two such ranges being without ambiguity when three points of one are assigned to correspond to three points of the other. This establishes a corresponding theory for related flat pencils of lines in a plane, all passing through the same point, and for related axial pencils of planes in space, all passing through the same line.