1 Euclidean geometry
We familiarise ourselves with the axioms of Euclidean geometry in the plane and derive some geometric implications, among them the existence of the parallel line. We briefly discuss the historical importance of the parallel axiom. The existence of the Cartesian model shows the consistency of the Euclidean axioms. The Cartesian model is used to investigate Euclidean trigonometry.
1.1 The axiomatic approach
Geometry is one of the oldest of all sciences. Remarkable geometric knowledge was already present in the advanced oriental cultures of the fifth–third centuries BCE. Practical problems from metrology, architecture, astronomy and navigation were considered on an abstract level and led to geometric laws. For instance, ...