The introduction of congruence. In Chapters V-VII we introduced the elliptic metric into real projective geometry by means of the “absolute polarity,” and observed the equivalence of two alternative definitions for a congruent transformation: a point-to-point transformation preserving distance, and a collineation permutable with the absolute polarity. It is quite easy to introduce the hyperbolic metric similarly (see §8.1). But in order to follow the historical development more closely, we prefer to reverse the process, introducing congruence into descriptive geometry as a second undefined relation, and stating its properties in the form of axioms. The propositions of Bolyai's “absolute geometry” can then be deduced in a straightforward manner. After imbedding the descriptive space in a real projective space by the method of Chapter VIII, we shall find a definite polarity which is permutable with every congruent transformation.

The relation of congruence applies initially to point-pairs, and we write AB ≡ CD to mean that the point-pair AB is congruent to the point-pair CD. But since every point-pair determines a unique segment, no confusion will be caused by reading the same formula as “the segment AB is congruent to the segment CD.”