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A Note on Absolute Geometry

Published online by Cambridge University Press:  20 November 2018

Roland Brossard*
Affiliation:
Université de Montréal
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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 We can verify that every axiom for the absolute geometry in [4] in a theorem in [3] where the similarity axiom has been replaced by the congruence axiom.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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Borsuk, K. and Szmielew, W., Foundations of geometry (Amsterdam 1960).Google Scholar
2. Brossard, R., Birkhoff's axioms for space geometry. Amer. Math. Monthly 71 (1964) 593-606.Google Scholar
3. Brossard, R., Metric axioms for space geometry. Amer. Math. Monthly 74 (1967) 777-788.Google Scholar
4. Coxeter, H.S.M., Introduction to geometry (John Wiley 1961).Google Scholar