By $EH$-geometry we understood the two-dimensional geometry which is dual to (Lobachevsky's) hyperbolic geometry. Without employing the duality principle, we have attempted a foundation of an order geometry, a geometry founded on incidence and order axioms. Basic to this geometry is the postulation of a non-empty set $\mathcal J$, the $C_a$, $C_{\tilde{a}}$ and $C_{\bar{a}}$-class which are subsets of $\mathcal J$, as well as two basic relations. The set $\mathcal J$ is then labeled as an $EH$-plane, and its elements are the points of the plane. The elements of the class $C_a$ are labeled as $EH$-plane lines, $C_{\tilde{a}}$-class elements are isotropic lines and $C_{\bar{a}}$-class elements are ideal line. The basic relation is two-member incidence relation $i\subset\mathcal JxC_a$, $i\subset\mathcal J xC_{\tilde{a}}$, $i\subset\mathcal JxC_{\bar{a}}$, which defines the point-line set relation, the point-isotropic line set relation and the point-ideal line set relation. Another basic relation is a four-member relation of separation of two point pairs which are incidence to one line. The present order geometry is founded on 8 incident axioms and 10 ordering axioms, its consistency being proven by the projective model.