ABSTRACT
Group theory is the best approach to geometry. But group theory involves "operations" and Plato did not approve of operations because operating clearly involved time, and truth—especially mathematical truth—was timeless. The relations between the different geometries can be set out in different ways. The sum of the angles of a triangle in Euclidean geometry is two right angles; in elliptical geometry it is always more than two right angles; in hyperbolic geometry it is always less. A similar relationship can be made out in terms of the axioms of parallels. Absolute and affine geometry are like projective geometry, and unlike elliptical, Euclidean and hyperbolic geometry, in not starting with a concept of congruence, of equality of distance and of angle, and having therefore no concept of a right angle or of similar triangles. Euclidean geometry is then a special case of affine geometry, having these concepts since it is a metrical geometry.
