Elliptic geometry in general. As we saw in §1.7, Klein's elliptic geometry is a metrical geometry in which two coplanar lines always have a single point of intersection. One method of approaching this geometry is to introduce an undefined relation of congruence, satisfying certain axioms such as the following:

5.11. From any point D on a given line, we can lay off two segments,CD and DE , each congruent to a given segment AB.

We can then develop a chain of propositions similar to Euclid I, 1-15, and conclude that all lines are finite and equal. A line being finite, any two points determine two “supplementary” segments. If these are equal, each is a “right” segment, and the two points are said to be “conjugate.” All the lines perpendicular to a given plane are found to concur at a definite point, conjugate to every point in the plane. Conversely, the locus of points conjugate to a given point is a plane. There is thus a definite correspondence between points and planes, of the kind that was called a uniform polarity in §3.8.

This brings us to the alternative treatment which is followed in the present book. Observing that every axiom of real projective geometry is valid in elliptic geometry, we simply adopt the axioms of §2.1, and agree to specialize a uniform polarity. This absolute polarity, having once been arbitrarily chosen, is kept fixed, and a congruent transformation is defined as a collineation which transforms this polarity into itself.