Axiomatization by Means of Coordinates

Abstract

Since we have imposed on Euclidean Geometry the duty of using the field R of real numbers as distance system, this is easiest to understand as a twodimensional vector space ε over ℝ. With the help of the scalar product the distance function is defined to be $\left\| {x,y} \right\| = \sqrt {\left( {x - y} \right)\left( {x - y} \right)}$ for arbitrary x,y ∈ ε.