A concise development of a consistent non-Euclidean geometry obtained by rejecting Euclid’s parallel postulate and working out its consequences: it shows that through a point not on a given line there are many lines that never meet the given line, introduces the angle of parallelism and related functions, derives new trigonometric relations for triangles in this geometry, and argues for the internal logical coherence and foundational significance of the alternative, hyperbolic conception of space.