A Note on Absolute Geometry

Abstract

Metric axioms have been given in [3] for space euclidean geometry. If we replace the "similarity axiom" by the "congruence axiom", where congruence is defined to be a similarity of ratio one, the resulting structure is absolute geometry. In order to show this we choose a suitable definition for absolute geometry. The P a s c h system of axioms, given in an improved formulation by H. S. M. Coxeter in [4], is particularly suitable; the primitive notions are points, betweenness relation, and congruence relation. We can verify that every axiom for the absolute geometry in [4] in a theorem in [3] where the similarity axiom has been replaced by the congruence axiom. The only case for which it is not obvious is axiom 15.15 in [4] which says that if ABC and A' B' C' are two triangles with BC ≡ B'C' CA ≡ C'A1, AB ≡ A ' B ', while D and D' are two further points such that [B, C, D] and [B', C' D'] and BD ≡ B' D', then AD ≡ A' D'. In that case we first prove that if two triangles ABC and A'B C are such that AB/A'B' ≡ BC/B'C' ≡ CA/C'A' ≡ 1 then they are congruent; a proof of this, independent of the similarity axiom, can be found in [2]. The proof of 15. 15 in [4] is then obvious. As every axiom in the weakened structure of [3] is a theorem of absolute geometry we have a definition for this geometry.